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</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="206" 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="1" left="60" top="6.75" width="349.5" height="14.25" align-x="60" align-y="18" 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 font-weight="normal">Гл.1, </inlineAttr> </f> <f family="Times New Roman" charset="204" size="12"> <inlineAttr font-weight="normal"> <c val="#000">§3, п.3. </c> </inlineAttr> </f> <f family="Times New Roman" charset="204" size="12"> <b> <c val="#000">Дискретная модель кратного интеграла.</c> </b> </f> </p> </text> </region> <region region-id="90" left="48" top="42.75" width="300.75" height="13.5" align-x="48" align-y="54" 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">Вычислить объем тела, ограниченного поверхностями:</f> </p> </text> </region> <region region-id="10" left="54" top="67.5" width="66" height="21" align-x="99.75" align-y="84" 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">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="28" left="144" top="67.5" width="92.25" height="21" align-x="189.75" align-y="84" 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">y2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="29" left="264" top="72.75" width="81" height="15.75" align-x="309.75" align-y="84" 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">y3</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="92" left="54" top="96.75" width="438.75" height="41.25" align-x="54" 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="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <b>Решение</b> </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12">Первые две поверхности являются цилиндрическими и оставляют на плоскости </f> <f family="Arial Cyr" charset="204" size="12">x0z следы, изображенные на<sp count="2"/>рис.1. Область, ограниченную<sp count="2"/>кривыми y11(z) и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">y22(z) на рис.1, можно принять за область интегрирования.</f> </p> </text> </region> <region region-id="64" left="60" top="163.5" width="60.75" height="21" align-x="100.5" align-y="180" 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">y11</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="63" left="168" top="163.5" width="87" height="21" align-x="208.5" align-y="180" 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">y22</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="95" left="306" top="166.5" width="39.75" height="18" align-x="320.25" align-y="180" 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:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="13" left="102" top="216" width="264" height="161.25" align-x="102" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="8"/> <rendering item-idref="9"/> </region> <region region-id="100" left="54" top="426.75" width="440.25" height="40.5" align-x="54" align-y="438" 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">Тело, объем которого требуется определить, имеет два варианта графического представления соответсвенно </f> <f family="Arial Cyr" charset="204" size="12">на рис. 2 и 4. Исходя из области интегрирования</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">рис.1, его объем определяется посредством двойного интеграла: </f> </p> </text> </region> <region region-id="105" left="66" top="495.75" width="145.5" height="44.25" align-x="83.25" align-y="522" 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">V</ml:id> <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:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:id xml:space="preserve">y11</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">y22</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="109" left="240" top="495.75" width="60.75" height="46.5" align-x="257.25" align-y="528" 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:id xml:space="preserve">V</ml:id> <ml:symResult> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:div/> <ml:real>32</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="110" left="324" top="516.75" width="65.25" height="15.75" align-x="339.75" align-y="528" 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:id xml:space="preserve">V</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>15.084944665313017</ml:real> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="127" left="54" top="570.75" width="435" height="54" align-x="54" align-y="582" 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">Таким образом, вычисление объема тела не представляет большой сложности,</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">если правильно построено само тело, определена область интегрирования и </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">выбрана система координат. Поэтому так важно дать правильную геометри-</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">ческую интерпретацию условиям задачи. Рассмотрим два варианта графики.</f> </p> </text> </region> <region region-id="51" left="48" top="660.75" width="114.75" height="13.5" align-x="48" align-y="672" 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">1-й вариант графики.</f> </p> </text> </region> <region region-id="48" left="42" top="726" width="361.5" height="352.5" align-x="222" align-y="726" 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="13" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">y1</ml:id> <ml:id xml:space="preserve">y2</ml:id> <ml:id xml:space="preserve">y3</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="14"/> </region> <region region-id="129" left="36" top="1104.75" width="469.5" height="40.5" align-x="36" align-y="1116" 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">На этом рисунке видны все ограничивающие тело поверхности. Именно из рас- смотрения этого рисунка можно сделать вывод о том какой интеграл проще записать</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">для определения объема тела. </f> </p> </text> </region> <region region-id="131" left="42" top="1164.75" width="114.75" height="13.5" align-x="42" align-y="1176" 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">2-й вариант графики.</f> </p> </text> </region> <region region-id="132" left="42" top="1188.75" width="459.75" height="41.25" align-x="42" align-y="1200" 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">Часто бывает удобнее представлять поверхность на рисунке в виде матрицы.<sp count="2"/>Для этого п</f> <f family="Arial Cyr" charset="204" size="12">ерейдем к дискретным переменным и зададим дискрет </f> <f family="Arial Cyr" charset="204" size="12"> <i>h </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и другие пара- метры</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">:</f> </p> </text> </region> <region region-id="165" left="48" top="1242.75" width="46.5" height="15.75" align-x="63" align-y="1254" show-border="false" show-highlight="true" 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">h</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="156" left="36" top="1272" width="464.25" height="30.75" align-x="36" align-y="1284" 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 независимые индекс </f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12"> и индексированную переменную </f> <f family="Arial Cyr" charset="204" size="12"> <i>z</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>m</sub> </f> <f family="Arial Cyr" charset="204" size="12">, предварительно</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">определив диапазон их изменения: </f> </p> </text> </region> <region region-id="157" left="42" top="1318.5" width="110.25" height="33.75" align-x="66" align-y="1338" 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">mh</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">a</ml:id> <ml:parens> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="164" left="174" top="1326.75" width="57" height="15.75" align-x="196.5" align-y="1338" 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:id xml:space="preserve">mh</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>141</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="159" left="264" top="1326.75" width="59.25" height="15.75" align-x="282" align-y="1338" 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:id xml:space="preserve">mh</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="160" left="342" top="1326.75" width="76.5" height="17.25" align-x="363.75" align-y="1338" 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">z</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="166" left="42" top="1368" width="395.25" height="14.25" align-x="42" align-y="1380" 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 </f> <f family="Arial Cyr" charset="204" size="12">диапазоны изменения</f> <f family="Arial Cyr" charset="204" size="12"> зависимых от </f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>индексов </f> <f family="Arial Cyr" charset="204" size="12"> <i>n</i> </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12">) и </f> <f family="Arial Cyr" charset="204" size="12"> <i>n</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12">):</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </p> </text> </region> <region region-id="178" left="36" top="1421.25" width="144.75" height="39" align-x="75.75" align-y="1446" 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">n1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="179" left="186" top="1434.75" width="72.75" height="15.75" align-x="230.25" align-y="1446" 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">n1</ml:id> <ml:id xml:space="preserve">mh</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>98</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="180" left="270" top="1421.25" width="164.25" height="39" align-x="309.75" align-y="1446" 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">n2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="181" left="438" top="1434.75" width="63.75" height="15.75" align-x="473.25" align-y="1446" 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">n2</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>99</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="182" left="36" top="1470" width="458.25" height="58.5" align-x="36" align-y="1482" 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">Каждый из этих индексов </f> <f family="Arial Cyr" charset="204" size="12"> <i>n</i> </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12">) и </f> <f family="Arial Cyr" charset="204" size="12"> <i>n</i> </f> <f family="Arial Cyr" charset="204" size="12">2</f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>m</i> </f> <f family="Arial Cyr" charset="204" size="12">) задает соответственно нижний и верхний</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">пределы изменения дискретной переменной y</f> <f family="Arial Cyr" charset="204" size="12"> <sub>n</sub> </f> <f family="Arial Cyr" charset="204" size="12">. Иллюстрацией этого может быть</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">рис.3. Теперь зададим проецирующие поверхности </f> <f family="Arial Cyr" charset="204" size="12"> <i>y</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> <i> <f family="Arial Cyr" charset="204" size="12">z</f> </i> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>и </f> <f family="Arial Cyr" charset="204" size="12"> <i>y</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</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>z</i> </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> в виде матриц,</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">например, так:</f> </p> </text> </region> <region region-id="184" left="72" top="1548.75" width="83.25" height="17.25" align-x="138.75" align-y="1560" 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">Q1</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="183" left="222" top="1548.75" width="83.25" height="17.25" align-x="288.75" align-y="1560" 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">Q2</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">n2</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="82" left="48" top="1596" width="268.5" height="210" align-x="181.5" align-y="1596" 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="26" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Q1</ml:id> <ml:id xml:space="preserve">Q2</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="27"/> </region> <region region-id="185" left="48" top="1836.75" width="470.25" height="41.25" align-x="48" 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">После этого перейдем к построению тела рис.4. В точках области интегрирования,</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">задаваемых поверхностями Q1 и Q2 или кривыми y11(z) и<sp count="2"/>y22(z), определим аппликату </f> <f family="Arial Cyr" charset="204" size="12"> <i>X</i> </f> <f family="Arial Cyr" charset="204" size="12">=4-</f> <f family="Arial Cyr" charset="204" size="12"> <i>y </i> </f> <f family="Arial Cyr" charset="204" size="12"> и запишем её в </f> <f family="Arial Cyr" charset="204" size="12">матрицу </f> <f family="Arial Cyr" charset="204" size="12"> <i>Н </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">по следующему алгоритму:</inlineAttr> </f> </p> </text> </region> <region region-id="85" left="72" top="1896.75" width="166.5" height="80.25" align-x="89.25" align-y="1908" 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">H</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">mh</ml:id> </ml:range> <ml:for> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">n2</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">H</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> </ml:for> <ml:id xml:space="preserve">H</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="194" left="48" top="2004.75" width="443.25" height="42" align-x="48" align-y="2016" 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">Этот алгоритм передает или соответствует процедуре интегрирования при опре-</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">делении объема </f> <f family="Arial Cyr" charset="204" size="12"> <i>V</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, но записанной в дискретном виде. Здесь оператор </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>for </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">играет</inlineAttr> </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 font-style="normal">роль интегралов, а индексы<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>n</i> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>и </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <i>m </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">-<sp count="2"/>их пределов интегрирования.</inlineAttr> </f> </p> </text> </region> <region region-id="192" left="54" top="2064" width="439.5" height="14.25" align-x="54" align-y="2076" 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">Далее строим тело рис.4, где аппликатой будет </f> <f family="Arial Cyr" charset="204" size="12"> <i>х,</i> </f> <f family="Arial Cyr" charset="204" size="12"> абсциссой </f> <f family="Arial Cyr" charset="204" size="12"> <i>у</i> </f> <f family="Arial Cyr" charset="204" size="12"> и ординатой </f> <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="191" left="84" top="2118" width="369.75" height="366" align-x="267.75" align-y="2118" 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="29" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">H</ml:id> </inputs> <outputs/> </component> <rendering item-idref="30"/> </region> <region region-id="204" left="54" top="2502.75" width="450" height="78" align-x="54" align-y="2514" 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">Если в приведенном выше алгоритме каждой аппликате поставить в соответствие</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">элементарную площадку </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">, на которой она восстановлена, то произведение </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>H</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>n,m</sub> </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <sup> <f family="Arial Cyr" charset="204" size="12">2</f> </sup> <f family="Arial Cyr" charset="204" size="12"> будет</f> <f family="Arial Cyr" charset="204" size="12"> определять элементарный объем. Сумма же этих элементарных</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">объемов даст объем тела. Это и будет дискретной моделью кратного интеграла. Убедимся в этом экспериментально.</f> </p> </text> </region> <region region-id="196" left="72" top="2598.75" width="177" height="129" align-x="89.25" align-y="2610" 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:program> <ml:localDefine> <ml:id xml:space="preserve">G</ml:id> <ml:real>0</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">mh</ml:id> </ml:range> <ml:for> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">n2</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">H</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:id xml:space="preserve">G</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">G</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">H</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> </ml:program> </ml:for> </ml:for> <ml:id xml:space="preserve">G</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="198" left="48" top="2742.75" width="65.25" height="15.75" align-x="63.75" align-y="2754" 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:id xml:space="preserve">G</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>15.234031999999884</ml:real> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="199" left="138" top="2742.75" width="65.25" height="15.75" align-x="153.75" align-y="2754" 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:id xml:space="preserve">V</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>15.084944665313017</ml:real> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="206" left="42" top="2766.75" width="451.5" height="27.75" align-x="42" align-y="2778" 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">Итак, погрешность в вычислениях составляет менее одного процента и зависит от</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">дискрета </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, что так же можно экспериментально проверить</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">.</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">iVBORw0KGgoAAAANSUhEUgAAAFgAAAAcCAYAAADlXHhOAAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAJHSURBVGhD7VgLsoMgDORcHIjzeBou 42F4EEADho8+UvvmxZlOO1Zj2CybjcrJwYqAYo0uwZ0AzEwCAVgAZkZgZXhrnFIqfoyFyMLg VQDvm9MJVBd+e5D1tgvAy/C11u0o2L5pp/QmAK8C+BInyIUAzAavVwktEsEHr3XGszdIhjQ5 BpStMS56CAF4ObxBGrKZeADw7jbtPV6ifzs7v0XUWcW1q+CM/ctMfWML1uw4QqObDxkWlkx0 D2Aw21zgpmzBZ2qH1zK/Dp4rwZZlfI7vWwDHxLK/w54PlYyRuRUwADJzIRfU4gaDxwBbE6eX 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