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\viewkind4\uc1\pard\qc\f0\fs18\'d2\'e8\'f2\'ee\'e2 \'ca.\'c2.\par }</center> <right>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="313" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="167" left="6" top="12.75" width="498" height="71.25" align-x="20.25" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <b> <inlineAttr underline="false" line-through="false">Пример типовой задачи вычисления тройного интеграла в Mathcad.</inlineAttr> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Вычислить </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">тройной интеграл от функции f(x,y,z)</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> по области V, ограниченной данными </inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">поверхностями.</f> </inlineAttr> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Дано: 1)<sp count="2"/> </inlineAttr> </f> <region region-id="179" left="56.25" top="54.75" width="63" height="15.75" align-x="104.25" align-y="66" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:id xml:space="preserve">z</ml:id> </ml:define> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ;</inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f size="12"> <sp count="12"/>2) </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">уравнения поверхностей, ограничивающих </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">область V</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> (см. рис.2):</inlineAttr> </f> </p> </text> </region> <region region-id="294" left="18" top="96.75" width="399" height="13.5" align-x="18" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">- укажем параметры, которые можно изменять в интерактивном режиме:</f> </p> </text> </region> <region region-id="301" left="24" top="120.75" width="30.75" height="15.75" align-x="38.25" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">b</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="300" left="78" top="120.75" width="39" height="15.75" align-x="91.5" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>0.5</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="299" left="144" top="120.75" width="30" height="15.75" align-x="157.5" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="285" left="12" top="145.5" width="108" height="21" align-x="57" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="302" left="150" top="150.75" width="61.5" height="15.75" align-x="195" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:id xml:space="preserve">b</ml:id> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="306" left="234" top="150.75" width="109.5" height="15.75" align-x="279" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:real>100</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="307" left="378" top="150.75" width="109.5" height="15.75" align-x="423" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z4</ml:id> <ml:boundVars> <ml:id 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size="12">цилиндрическими</f> <f family="Arial Cyr" charset="204" size="12"> поверхностями (плоскостями) х=a и у=g, которые в дальнейшем будут использоваться только в графи-</f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">ческих построениях.</f> </p> </text> </region> <region region-id="71" left="6" top="228.75" width="536.25" height="67.5" align-x="21.75" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Для определения и растановки пределов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">интегрирования в тройном интеграле </f> <f family="Arial Cyr" charset="204" size="12">в декар-</f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">товой системе координат</f> <f family="Arial Cyr" charset="204" size="12"> необходимо последовательно провести проецирование области </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">V в</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> область интегрирования D на плоскости и далее D - спроецировать на одну из коорди-</inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">натных осей. Здесь удобно </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> получить на плоскости х0у. 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<ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="157" left="138" top="336.75" width="27" height="15.75" align-x="150" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </math> <rendering item-idref="12"/> </region> <region region-id="153" left="24" top="357" width="200.25" height="84" align-x="89.25" align-y="402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:Find auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> <ml:symResult> <ml:matrix rows="3" cols="2"> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>3</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:real>1</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>3</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="13"/> </region> <region region-id="162" left="12" top="480.75" width="491.25" height="105" align-x="24.75" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Каждый из полученных вектор-столбцов является решением исходной системы </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">уравне</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ний. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Таким образом, из </inlineAttr> </f> <region region-id="163" left="177.75" top="495" width="87.75" height="23.25" align-x="190.5" align-y="513.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="14"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> следует, что</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> поверхности пересека-</inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ются по окруж</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ности радиусом<sp count="2"/> </inlineAttr> </f> <region region-id="309" left="180.75" top="519" width="34.5" height="18" align-x="192.75" align-y="532.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">. Покажем её на рис.1. Кроме этого там же </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">укажем<sp count="2"/>другие линии гра</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ницы области D. Итак, </inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">область D на плоскости x0y полностью определяется следующими линиями:</f> </inlineAttr> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <region region-id="168" left="15" top="564.75" width="75.75" height="21" align-x="57" align-y="581.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="16"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/> </inlineAttr> </f> <region region-id="169" left="100.5" top="570" width="27" height="15.75" align-x="113.25" align-y="581.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">g</ml:id> </ml:apply> </math> <rendering item-idref="17"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и </inlineAttr> </f> <region region-id="170" left="143.25" top="570" width="27" height="15.75" align-x="156" align-y="581.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </math> <rendering item-idref="18"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> </p> </text> </region> <region region-id="310" left="18" top="594.75" width="476.25" height="27.75" align-x="18" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Введем обозначение радиуса окружности, по которой пересекаются поверхности</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>z</i> </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> <i>y</i> </f> <f family="Arial Cyr" charset="204" size="12">)<sp count="2"/>и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>z2</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <i> <f family="Arial Cyr" charset="204" size="12">x</f> </i> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> <i>y</i> </f> <f family="Arial Cyr" charset="204" size="12">)</f> </p> </text> </region> <region region-id="311" left="48" top="634.5" width="60" height="18" align-x="63.75" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="313" left="54" top="708" width="205.5" height="205.5" align-x="54" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="20"/> <rendering item-idref="21"/> </region> <region region-id="172" left="42" top="954" width="442.5" height="28.5" align-x="42" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Из рис. 1 понятно какие пределы следует взять по переменным </f> <f family="Arial Cyr" charset="204" size="12"> <i>х</i> </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> <i>у. </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">Рис.2 дает </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">общее </inlineAttr> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">представление об области V и определяет пределы интегрирования по </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <i>z</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. </inlineAttr> </f> </p> </text> </region> <region region-id="143" left="144" top="1038" width="285" height="333" align-x="285.75" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="22" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z1</ml:id> <ml:id xml:space="preserve">z2</ml:id> <ml:id xml:space="preserve">z3</ml:id> <ml:id xml:space="preserve">z4</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="23"/> </region> <region region-id="110" left="6" top="1410.75" width="475.5" height="13.5" align-x="27.75" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Теперь проведем вычисление тройного интеграла. </inlineAttr> </f> </p> </text> </region> <region region-id="178" left="30" top="1437" width="234.75" height="51" align-x="57.75" align-y="1470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">J</ml:id> <ml:boundVars> <ml:id xml:space="preserve">a</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">R</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="180" left="282" top="1450.5" width="195" height="33.75" align-x="309.75" align-y="1470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">J</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>4</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3.6002752278078960832</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="25"/> </region> <region region-id="183" left="36" top="1522.5" width="204.75" height="33.75" align-x="73.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">J</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>4</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>1.5104128940588061835</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="26"/> </region> <region region-id="186" left="294" top="1522.5" width="60" height="33.75" align-x="322.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">J</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>4</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="27"/> </region> <region region-id="197" left="402" top="1530.75" width="69.75" height="15.75" align-x="429" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">J</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.4683082221231167</ml:real> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="187" left="36" top="1572.75" width="441.75" height="40.5" align-x="36" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Очевидно, что </f> <f family="Arial Cyr" charset="204" size="12">вычисление этого тройного интеграла проще было бы провести в </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">цилиндри</f> <f family="Arial Cyr" charset="204" size="12">ческой системе координат, переход в которую осуществляется по фор-</f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">мулам:</f> </p> </text> </region> <region region-id="188" left="48" top="1626" width="63.75" height="16.5" align-x="60.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">r</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="29"/> </region> <region region-id="189" left="138" top="1626" width="60" height="16.5" align-x="150.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">r</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <region region-id="190" left="216" top="1626.75" width="25.5" height="15.75" align-x="228" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="191" left="42" top="1656" width="460.5" height="41.25" align-x="42" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">При переходе в цилиндрическую систему координат якобиан равен </f> <f family="Arial Cyr" charset="204" size="12"> <i>r</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Поэтому, </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">исходя из </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">рис.1,2, расстановка пределов в тройном интеграле в этой </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">системе </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">коор</f> <f family="Arial Cyr" charset="204" size="12">динат</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> будет следующей:<sp count="2"/> </inlineAttr> </f> </p> </text> </region> <region region-id="193" left="48" top="1744.5" width="218.25" height="66" align-x="81" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Jc</ml:id> <ml:boundVars> <ml:id xml:space="preserve">a</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:div/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">R</ml:id> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">acos</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="292" left="30" top="1836.75" width="159.75" height="13.5" align-x="30" align-y="1848" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Получим значение интеграла:</f> </p> </text> </region> <region region-id="279" left="222" top="1836.75" width="166.5" height="15.75" align-x="255" align-y="1848" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Jc</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:symResult> <ml:real>3.2942723816876715735</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="33"/> </region> <region region-id="280" left="426" top="1828.5" width="65.25" height="33.75" align-x="459.75" align-y="1848" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Jc</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>4</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="34"/> </region> <region region-id="205" left="54" top="1878.75" width="457.5" height="27" align-x="54" align-y="1890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Итак, результаты полученных вычислений полностью совпадают. Представим </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">пошаговое вычисление тройного интеграла:</f> </p> </text> </region> <region region-id="201" left="78" top="1930.5" width="212.25" height="49.5" align-x="115.5" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Jc1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>15</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="202" left="318" top="1942.5" width="137.25" height="33.75" align-x="355.5" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Jc1</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>15</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="36"/> </region> <region region-id="206" left="78" top="2001.75" width="387" height="60.75" align-x="132" align-y="2028" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Jc2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Jc1</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:div/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">R</ml:id> </ml:bounds> </ml:apply> <ml:symResult> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>12</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>45</ml:real> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>12</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>6</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="281" left="42" top="2112.75" width="477" height="27" align-x="42" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Далее для каждого слагаемого из правой части можно записать интег</f> <f family="Arial Cyr" charset="204" size="12">рал, который </f> <f family="Arial Cyr" charset="204" size="12">вычисляется по рекуррентной формуле (см. Ефимов, т.1, стр.282):</f> </p> </text> </region> <region region-id="218" left="84" top="2153.25" width="166.5" height="65.25" align-x="132.75" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:integral/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="38"/> </region> <region region-id="219" left="282" top="2164.5" width="85.5" height="33.75" align-x="356.25" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">m</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:sequence> </math> <rendering item-idref="39"/> </region> <region region-id="260" left="78" top="2238.75" width="72.75" height="13.5" align-x="78" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Где, полагая </f> </p> </text> </region> <region region-id="259" left="162" top="2238.75" width="33.75" height="17.25" align-x="179.25" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">I</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="258" left="216" top="2238.75" width="48.75" height="15.75" align-x="232.5" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="262" left="84" top="2271.75" width="297.75" height="40.5" align-x="133.5" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_variable mc_φ mc_φ value_zone " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="263" left="84" top="2334.75" width="47.25" height="13.5" align-x="84" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">получим:</f> </p> </text> </region> <region region-id="244" left="132" top="2369.25" width="276.75" height="126.75" align-x="181.5" align-y="2436" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>5</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>8</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="43"/> </region> <region region-id="265" left="72" top="2550" width="423.75" height="14.25" align-x="72" align-y="2562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Или же сразу можно провести вычисление заключительного интеграла</f> <f family="Arial Cyr" charset="204" size="12"> <i> J</i> </f> <f family="Arial Cyr" charset="204" size="12"> <i>c</i> </f> <f family="Arial Cyr" charset="204" size="12">3(</f> <f family="Arial Cyr" charset="204" size="12"> <i>a</i> </f> <f family="Arial Cyr" charset="204" size="12">):</f> </p> </text> </region> <region region-id="284" left="6" top="2584.5" width="507" height="55.5" align-x="45" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Jc3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">a</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Jc2</ml:id> <ml:sequence> <ml:id xml:space="preserve">φ</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">acos</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:bounds> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> 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<ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>45</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="210" left="90" top="2682.75" width="182.25" height="15.75" align-x="138.75" align-y="2694" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Jc3</ml:id> <ml:real>0.5</ml:real> </ml:apply> <ml:symResult> <ml:real>3.2942723816876715738</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="45"/> </region> <region region-id="212" left="312" top="2674.5" width="71.25" height="33.75" align-x="351.75" align-y="2694" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">Jc3</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>9</ml:real> <ml:real>4</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="46"/> </region> <region region-id="293" left="54" top="2718.75" width="450" height="40.5" align-x="54" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Итак, компьютерное изложение метода вычисления тройного интеграла получено.</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Очевидно, что оно позволяет избежать рутинных вычислений, экономя тем самым время на осмысление метода вычисления. </f> </p> </text> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/ GQZExBiHSVQb3OPjcnep3PO6Oz+q0tURyZljYrVzVzu1aIHplcX2cD8VdWPtmxcFM1ym7Z9K q2r5Stz79lprtVE2w2y4ZZvj8RBn/7u/8m5dueK0zANd2RZMeUJs2Ytitlj6lAO2BtGUDdvQ ixOCIQ8RgfmssOoII6muMhA9YQSOpUfiCRiN98FofABG40P9F8RIGYixMAJ/AQAA//8DAMuS Xhh8AQAA</item> <item item-id="2">iVBORw0KGgoAAAANSUhEUgAAAFQAAAAVCAYAAADYb8kIAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA AUlJREFUWEftmOEOwiAMhOf7P7RuixDGoHdXuoRkkPhDV+H4Sq/o57uPbY04AgfQNeIIbHFT rZnOal8YYgksoLE87yd0d+ejSZ2verQ+C9bjmm4mXRdqtbDy/UyiW9Rn0WcCTcJnEYuO7ww6 Xwn0SfAZaOmdqNRrn7V8t+XDpUcr3/3/CMke3/L6J2H1rObCrgxSGpHlt2ppKhCYZKP16z3X h0lJcq0Hlry12fRMAZJOGUokgtJbU9WC1mGfZxZoY0gget4T5E1GKyFoDywUb9zllCIxT5zQ 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