MIT_单变量微积分_20

雨点打透心脏的1/2处 2022-03-06 10:38 215阅读 0赞

### 1.壳层法、圆盘法求体积 ###

`Ex` 利用切片球体积。

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 V = ∫ A ( x ) d x V=\\int A(x)dx V=∫A(x)dx

`Ex:`旋转立方体,绕x轴旋转(圆盘法)。  
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 d v = π y 2 d x dv=\\pi y^2dx dv=πy2dx

`Ex:`半径为 a a a球的体积。  
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 圆 盘 法 ： d y = π y 2 d x 圆 ： ( x − a ) 2 + y 2 = a 2 y 2 = a 2 − ( x − a ) 2 = 2 a x − x 2 V = ∫ 0 2 a π ( 2 a x − x 2 ) d x = π ( a x 2 − x 3 3 ) ∣ 0 2 a = π ( 4 a 3 − 8 a 3 3 ) = 4 3 π a 3 圆盘法：dy=\\pi y^2dx\\\\ 圆：(x-a)^2+y^2=a^2\\\\ y^2=a^2-(x-a)^2\\\\ =2ax-x^2\\\\ V=\\int\_0^\{2a\}\\pi(2ax-x^2)dx\\\\ =\\pi(ax^2-\\frac\{x^3\}\{3\})|\_0^\{2a\}\\\\ =\\pi(4a^3-\\frac\{8a^3\}\{3\})\\\\ =\\frac\{4\}\{3\}\\pi a^3 圆盘法：dy=πy2dx圆：(x−a)2\+y2=a2y2=a2−(x−a)2=2ax−x2V=∫02aπ(2ax−x2)dx=π(ax2−3x3)∣02a=π(4a3−38a3)=34πa3  
`Ex:`如图所示，类似于坩埚，绕y轴旋转(壳层法)。  
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 d v = ( 2 π x ) ( a − y ) d x = 2 π x ( a − x 2 ) d x = 2 π ( a x − x 3 ) d x V = ∫ 0 a 2 π ( a x − x 3 ) d x = ( π a x 2 − 2 π x 4 4 ) ∣ 0 a = π a 2 − π a 2 2 = π 2 a 2 dv=(2\\pi x)(a-y)dx\\\\ =2\\pi x(a-x^2)dx\\\\ =2\\pi (ax-x^3)dx\\\\ V=\\int\_0^\{\\sqrt\{a\}\}2\\pi(ax-x^3)dx\\\\ =(\\pi ax^2-2\\pi\\frac\{x^4\}\{4\})|\_0^\{\\sqrt\{a\}\}\\\\ =\\pi a^2-\\frac\{\\pi a^2\}\{2\}\\\\ =\\frac\{\\pi\}\{2\}a^2 dv=(2πx)(a−y)dx=2πx(a−x2)dx=2π(ax−x3)dxV=∫0a2π(ax−x3)dx=(πax2−2π4x4)∣0a=πa2−2πa2=2πa2

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