设D:x^2+y^2

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设D:x^2+y^2
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设D:x^2+y^2
设D:x^2+y^2

设D:x^2+y^2
由积分的几何意义,这个积分是半径为|a|的球的上半部分体积,有球的体积公式,
1/2*4/3π |a|^3=π
a=±(3/2)^(1/3)

化为极坐标:x = r*cosθ,y = r*sinθ(其中r <= a)
所以∫∫(√a^2-x^2-y^2)dxdy
= ∫∫(√a^2-r^2)rdrdθ(r从0积到a,θ从0积到2π)
= 2π * ∫(√a^2-r^2)rdr(r从0积到a)
(令r = a*cosα)
= 2π * ∫a*sinα*a*cosα d(a*cosα)(α从π/2积...

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化为极坐标:x = r*cosθ,y = r*sinθ(其中r <= a)
所以∫∫(√a^2-x^2-y^2)dxdy
= ∫∫(√a^2-r^2)rdrdθ(r从0积到a,θ从0积到2π)
= 2π * ∫(√a^2-r^2)rdr(r从0积到a)
(令r = a*cosα)
= 2π * ∫a*sinα*a*cosα d(a*cosα)(α从π/2积到0)
= -2π * a^3 * ∫(sinα)^2*cosα dα(α从π/2积到0)
= -2π * a^3 * ∫(sinα)^2 d(sinα)(α从π/2积到0)
= -2π/3 * a^3 * ∫d[(sinα)^3](α从π/2积到0)
= -2π/3 * a^3 * (0 - 1)
= 2π/3 * a^3

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