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# convert the integral 積分_-2~2 積

convert the integral 積分_-2~2 積_-(4-x^2)^1/2~(4-x^2)^1/2

積_(x^2+y^2)^1/2~2 dzdydx to an equivalent integral in cylindrical

coordinates

### 1 Answer

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- mathmanliuLv 71 decade agoFavorite Answer
原積分區域為:

-2<= x <= 2

-√(4-x^2) <= y <= √(4-x^2)

√(x^2+y^2) <= z <= 2

x, y形成之區域為平面上整個圓, 圓心(0,0), 半徑 2

因此將 xy改為極坐標, z不變(即為圓柱坐標)

原積分=∫[0~2π]∫[0~2]∫[r ~ 2] r dz dr dθ

=∫[0~2π]∫[0~2] (2-r)r dr dθ

=∫[0~2π] (4/3) dθ

= 8π/3

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