題目 Problem 9 (10%). Compute the integral ∫01∫−x−x2x−x2x2+y2 dy dx.\int_{0}^{1}\int_{-\sqrt{x-x^2}}^{\sqrt{x-x^2}}\sqrt{x^2+y^2}\,dy\,dx.∫01∫−x−x2x−x2x2+y2dydx. 解答 解法一 區域是以 (1/2,0)(1/2,0)(1/2,0) 為圓心、半徑 1/21/21/2 的圓盤；極座標下為 −π/2≤θ≤π/2-\pi/2\le\theta\le\pi/2−π/2≤θ≤π/2、0≤r≤cosθ0\le r\le\cos\theta0≤r≤cosθ。因此 ∫−π/2π/2∫0cosθr2 dr dθ=49.\int_{-\pi/2}^{\pi/2}\int_0^{\cos\theta}r^2\,\mathrm{d}r\,\mathrm{d}\theta =\boxed{\frac49}.∫−π/2π/2∫0cosθr2drdθ=94.
