x2+y2=2azz=xx^2+y^2=2az \qquad z=xx2+y2=2azz=x x=rsinθy=rsinθx=r\sin\theta \quad y=r\sin\thetax=rsinθy=rsinθ とおくと 0≤r≤2acosθ−π2≤θ≤π20\leq r\leq 2a\cos\theta \quad -\dfrac{\pi}{2}\leq\theta\leq\dfrac{\pi}{2}0≤r≤2acosθ−2π≤θ≤2π ∬Dx−12a(x2+y2) dxdy=∫−π2π2∫02acosθ(rcosθ−12ar2)r dr dθ=∫−π2π2[13r3cosθ−18ar4]02acosθdθ\iint_D x-\frac{1}{2a}(x^2+y^2)\,dxdy=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_0^{2a\cos\theta}\left(r\cos\theta-\frac{1}{2a}r^2\right)r\,dr\,d\theta=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[\frac{1}{3}r^3\cos\theta-\frac{1}{8a}r^4\right]_0^{2a\cos\theta}d\theta∬Dx−2a1(x2+y2)dxdy=∫−2π2π∫02acosθ(rcosθ−2a1r2)rdrdθ=∫−2π2π[31r3cosθ−8a1r4]02acosθdθ =∫−π2π283a3cos4θ−2a3cos4θ dθ=2∫0π283a3cos4θ−2a3cos4θ dθ=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{8}{3}a^3\cos^4\theta-2a^3\cos^4\theta\,d\theta=2\int_0^{\frac{\pi}{2}}\frac{8}{3}a^3\cos^4\theta-2a^3\cos^4\theta\,d\theta=∫−2π2π38a3cos4θ−2a3cos4θdθ=2∫02π38a3cos4θ−2a3cos4θdθ =2(83a3−2a3)(34⋅12⋅π2)=π4a3=2\left(\frac{8}{3}a^3-2a^3\right)\left(\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}\right)=\frac{\pi}{4}a^3=2(38a3−2a3)(43⋅21⋅2π)=4πa3