jack asked in 科學數學 · 1 decade ago

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

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  • 1 decade ago
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    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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