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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="185" 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="82" left="42" top="6.75" width="456.75" height="36" align-x="42" 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"> <f family="Arial Cyr" charset="204" size="12">Задачи из Ефимова. №8.85. Вычислить объём тела.</f> <f family="Arial Cyr" charset="204" size="9"> Примечание </f> <f family="Arial Cyr" charset="204" size="9">. Некоторые данные задачи изменены исключительно с целью лучшего графического изображения поверхностей.</f> </text> </region> <region region-id="83" left="42" top="48.75" width="303.75" height="13.5" align-x="42" align-y="60" 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"> <f family="Arial Cyr" charset="204" size="12">Выберем в интерактивном режиме параметры задачи: </f> </text> </region> <region region-id="59" left="54" top="72.75" width="30" height="15.75" align-x="67.5" 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:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="58" left="114" top="72.75" width="42.75" height="15.75" align-x="131.25" 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:id xml:space="preserve">ra</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="60" left="42" top="96.75" width="462.75" height="13.5" align-x="42" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем поверхности, ограничивающие тело, объем которого требуется вычислить.</f> </text> </region> <region region-id="77" left="48" top="125.25" width="86.25" height="39" align-x="87" 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</ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="71" left="468" top="138.75" width="15.75" height="15.75" align-x="474.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="7"/> </region> <region region-id="81" left="42" top="174.75" width="473.25" height="41.25" align-x="42" align-y="186" 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"> <f family="Arial Cyr" charset="204" size="12">Чтобы представить пространственное изображение тела рис.1, запишем цилиндр и плоскость из (1), кроме </f> <f family="Arial Cyr" charset="204" size="12">гиперболического 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xmlns="http://schemas.mathsoft.com/provenance10">64932d8165938103c4f5d49116b73dab</contentHash> <ml:define> <ml:id xml:space="preserve">mesh</ml:id> <ml:real>40</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="11"> <element-image-map> <box left="1.5" top="0.75" width="52.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="70" left="156" top="261.75" width="258" height="34.5" align-x="178.5" align-y="282" 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">Nn</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">X1</ml:id> <ml:id xml:space="preserve">Y1</ml:id> <ml:id xml:space="preserve">Z1</ml:id> <ml:real>-4</ml:real> <ml:real>4</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">mesh</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="84" left="42" top="306" width="467.25" height="42.75" align-x="42" align-y="318" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем плоскость в виде круга </f> <f family="Arial Cyr" charset="204" size="12"> N </f> <f family="Arial Cyr" charset="204" size="12">5 в параметрическом виде, используя (необя-</f> <f family="Arial Cyr" charset="204" size="12">зательно) коэффициент масштабирования </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">, который здесь определим как </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12">diapazon</f> <f family="Arial Cyr" charset="204" size="12">/</f> <f family="Arial Cyr" charset="204" size="12"> ra </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="97" left="42" top="364.5" width="36.75" height="33.75" align-x="56.25" align-y="384" 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">k</ml:id> <ml:apply> <ml:div/> <ml:real>5</ml:real> <ml:id xml:space="preserve">ra</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="98" left="114" top="364.5" width="101.25" 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xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="101" left="48" top="414.75" width="182.25" height="15.75" align-x="71.25" align-y="426" 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">N5</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">X5</ml:id> <ml:id xml:space="preserve">Y5</ml:id> <ml:id xml:space="preserve">Z5</ml:id> <ml:id xml:space="preserve">mesh</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="103" left="30" top="456" width="414.75" height="230.25" align-x="236.25" align-y="456" 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="18" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">Nn</ml:id> <ml:id xml:space="preserve">N5</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="19"/> </region> <region region-id="107" left="474" top="666.75" width="30" height="13.5" align-x="474" align-y="678" 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"> <f family="Arial Cyr" charset="204" size="12">Рис.1</f> </text> </region> <region region-id="24" left="36" top="708.75" width="474.75" height="27.75" align-x="36" align-y="720" 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"> <f family="Arial Cyr" charset="204" size="12">Теперь можно вычислить объём тела: </f> <f family="Arial Cyr" charset="204" size="12">1) в декартовой системе координат (здесь </f> <f family="Arial Cyr" charset="204" size="12"> с </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12">)</f> </text> </region> <region region-id="16" left="12" top="768.75" width="486" height="108.75" align-x="388.5" align-y="822" 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:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>8</ml:real> 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</ml:bounds> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>8</ml:real> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <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:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </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">c</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:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <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:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">csgn</ml:id> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="20"/> </region> <region region-id="21" left="30" top="930.75" width="220.5" height="27.75" align-x="30" align-y="942" 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"> <f family="Arial Cyr" charset="204" size="12">2) в полярной системе координат </f> <f family="Arial Cyr" charset="204" size="12">(здесь </f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="102" left="300" top="899.25" width="185.25" height="77.25" align-x="450.75" align-y="942" 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:mult style="default"/> <ml:real>4</ml:real> <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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> <ml:symResult> <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">b</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="21"/> </region> <region region-id="108" left="36" top="1002" width="485.25" height="41.25" align-x="36" align-y="1014" 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"> <f family="Arial Cyr" charset="204" size="12"> Приложение </f> <f family="Arial Cyr" charset="204" size="12">. Построим пересечение двух взаимно перпендикулярных цилиндров, используя ранее приведенные зависимости. Введём константы r2<sp count="2"/>и q. Первая</f> <f family="Arial Cyr" charset="204" size="12">характеризует диаметр цилиндра, а вторая - его высоту.</f> </text> </region> <region region-id="115" left="36" top="1056.75" width="34.5" height="15.75" align-x="54" align-y="1068" 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">r2</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="118" left="90" top="1056.75" width="30" height="15.75" align-x="103.5" align-y="1068" 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">h</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="132" left="144" top="1056.75" width="72.75" height="13.5" align-x="144" align-y="1068" 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"> <f family="Arial Cyr" charset="204" size="12">Здесь -h<r<h </f> </text> </region> <region region-id="166" left="12" top="1086" width="107.25" height="14.25" align-x="12" align-y="1098" 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"> <f family="Arial Cyr" charset="204" size="12"> Цилиндр с осью </f> <f family="Arial Cyr" charset="204" size="12"> 0Z </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="167" left="126" top="1086.75" width="102" height="15.75" align-x="173.25" align-y="1098" show-border="false" show-highlight="false" 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xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">r2</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="169" left="372" top="1078.5" width="142.5" height="33.75" align-x="417.75" align-y="1098" 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">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">r</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <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:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="172" left="126" top="1122.75" width="54" height="15.75" align-x="157.5" align-y="1134" 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="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">mesh</ml:id> <ml:real>40</ml:real> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="173" left="228" top="1122" width="247.5" height="16.5" align-x="249.75" align-y="1134" show-border="false" show-highlight="false" 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В системах компьютерной математики пространственные </f> <f family="Times New Roman Cyr" charset="204">графики строятся в правой системе координат, предполагающей по умолчанию следу</f> <f family="Times New Roman Cyr" charset="204">ющую последовательность осей: абсцисса, ордината и аппликата. Какие имена присва</f> <f family="Times New Roman Cyr" charset="204">ивать этим осям - не имеет значения. 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</ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="71" left="468" top="138.75" width="15.75" height="15.75" align-x="474.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="7"/> </region> <region region-id="81" left="42" top="174.75" width="473.25" height="41.25" align-x="42" align-y="186" 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"> <f family="Arial Cyr" charset="204" size="12">Чтобы представить пространственное изображение тела рис.1, запишем цилиндр и плоскость из (1), кроме </f> <f family="Arial Cyr" charset="204" size="12">гиперболического параболоида,</f> <f family="Arial Cyr" charset="204" size="12"> в параметрическом виде. </f> <f family="Arial Cyr" charset="204" size="12">Начнём с цилиндра </f> <f family="Arial Cyr" charset="204" size="12"> Nn </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="64" left="48" top="228.75" width="101.25" height="15.75" align-x="95.25" align-y="240" 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">X1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ra</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="68" left="192" top="228.75" width="97.5" height="15.75" align-x="239.25" align-y="240" 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="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ra</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="67" left="330" top="228.75" width="60" height="15.75" align-x="375.75" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="e7b06b53-5473-4096-bdf4-b51134ca75df" version-id="47057914-ed36-47de-8476-7cb4bfb85e4f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="23" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\СлушателиФПКП\Графика3D\ПострЦилиндрПоверхностей.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="e7b06b53-5473-4096-bdf4-b51134ca75df" version-id="47057914-ed36-47de-8476-7cb4bfb85e4f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="23" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\СлушателиФПКП\Графика3D\ПострЦилиндрПоверхностей.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">27153644054f95ce6f5cbb9a5bb48aa5</contentHash> <ml:define> <ml:function> <ml:id xml:space="preserve">Z1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:id xml:space="preserve">r</ml:id> </ml:define> </ml:provenance> </math> <rendering item-idref="10"> <element-image-map> <box left="1.5" top="0.75" width="58.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="39" left="54" top="270.75" width="54" height="15.75" align-x="85.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="23daf2e5-d6b9-41a9-818e-10680ba71a52" version-id="8afc3806-e1b9-42c9-b644-6ff32c151578" branch-id="00000000-0000-0000-0000-000000000000" revision-num="24" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\СлушателиФПКП\Графика3D\ПараметричПов.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="23daf2e5-d6b9-41a9-818e-10680ba71a52" version-id="8afc3806-e1b9-42c9-b644-6ff32c151578" branch-id="00000000-0000-0000-0000-000000000000" revision-num="24" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\СлушателиФПКП\Графика3D\ПараметричПов.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">64932d8165938103c4f5d49116b73dab</contentHash> <ml:define> <ml:id xml:space="preserve">mesh</ml:id> <ml:real>40</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="11"> <element-image-map> <box left="1.5" top="0.75" width="52.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="70" left="156" top="261.75" width="258" height="34.5" align-x="178.5" align-y="282" 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">Nn</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">X1</ml:id> <ml:id xml:space="preserve">Y1</ml:id> <ml:id xml:space="preserve">Z1</ml:id> <ml:real>-4</ml:real> <ml:real>4</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">mesh</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="84" left="42" top="306" width="467.25" height="42.75" align-x="42" align-y="318" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем плоскость в виде круга </f> <f family="Arial Cyr" charset="204" size="12"> N </f> <f family="Arial Cyr" charset="204" size="12">5 в параметрическом виде, используя (необя-</f> <f family="Arial Cyr" charset="204" size="12">зательно) коэффициент масштабирования </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">, который здесь определим как </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12">diapazon</f> <f family="Arial Cyr" charset="204" size="12">/</f> <f family="Arial Cyr" charset="204" size="12"> ra </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="97" left="42" top="364.5" width="36.75" height="33.75" align-x="56.25" align-y="384" 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">k</ml:id> <ml:apply> <ml:div/> <ml:real>5</ml:real> <ml:id xml:space="preserve">ra</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="98" left="114" top="364.5" width="101.25" 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disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Y5</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="100" left="366" top="372.75" width="62.25" height="15.75" align-x="411.75" align-y="384" 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">Z5</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:boundVars> </ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="101" left="48" top="414.75" width="182.25" height="15.75" align-x="71.25" align-y="426" 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">N5</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">X5</ml:id> <ml:id xml:space="preserve">Y5</ml:id> <ml:id xml:space="preserve">Z5</ml:id> <ml:id xml:space="preserve">mesh</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="103" left="30" top="456" width="414.75" height="230.25" align-x="236.25" align-y="456" 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="18" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">Nn</ml:id> <ml:id xml:space="preserve">N5</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="19"/> </region> <region region-id="107" left="474" top="666.75" width="30" height="13.5" align-x="474" align-y="678" 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"> <f family="Arial Cyr" charset="204" size="12">Рис.1</f> </text> </region> <region region-id="24" left="36" top="708.75" width="474.75" height="27.75" align-x="36" align-y="720" 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"> <f family="Arial Cyr" charset="204" size="12">Теперь можно вычислить объём тела: </f> <f family="Arial Cyr" charset="204" size="12">1) в декартовой системе координат (здесь </f> <f family="Arial Cyr" charset="204" size="12"> с </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12">)</f> </text> </region> <region region-id="16" left="12" top="768.75" width="486" height="108.75" align-x="388.5" align-y="822" 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:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>8</ml:real> 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</ml:apply> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <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:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">csgn</ml:id> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="20"/> </region> <region region-id="21" left="30" top="930.75" width="220.5" height="27.75" align-x="30" align-y="942" 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"> <f family="Arial Cyr" charset="204" size="12">2) в полярной системе координат </f> <f family="Arial Cyr" charset="204" size="12">(здесь </f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="102" left="300" top="899.25" width="185.25" height="77.25" align-x="450.75" align-y="942" 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use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Приложение </f> <f family="Arial Cyr" charset="204" size="12">. Построим пересечение двух взаимно перпендикулярных цилиндров, используя ранее приведенные зависимости. Введём константы r2<sp count="2"/>и q. Первая</f> <f family="Arial Cyr" charset="204" size="12">характеризует диаметр цилиндра, а вторая - его высоту.</f> </text> </region> <region region-id="115" left="36" top="1056.75" width="34.5" height="15.75" align-x="54" align-y="1068" 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">r2</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="118" left="90" top="1056.75" width="30" height="15.75" align-x="103.5" align-y="1068" 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">h</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="23"/> 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В системах компьютерной математики пространственные </f> <f family="Times New Roman Cyr" charset="204">графики строятся в правой системе координат, предполагающей по умолчанию следу</f> <f family="Times New Roman Cyr" charset="204">ющую последовательность осей: абсцисса, ордината и аппликата. Какие имена присва</f> <f family="Times New Roman Cyr" charset="204">ивать этим осям - не имеет значения. 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