z=a2−x2−y2z = \sqrt{a^2 - x^2 - y^2}z=a2−x2−y2 x=rcosθx = r\cos\thetax=rcosθ y=rsinθy = r\sin\thetay=rsinθ とおくと 0≤r≤b0 \le r \le b0≤r≤b 0≤θ≤π20 \le \theta \le \dfrac{\pi}{2}0≤θ≤2π として 8∫0π2∫0ba2−r2⋅r dr dθ=8∫0π2[−13(a2−r2)3]0bdθ=8∫0π2−13((a2−b2)3−a3)dθ8\int_0^{\frac{\pi}{2}}\int_0^b \sqrt{a^2 - r^2} \cdot r\,dr\,d\theta = 8\int_0^{\frac{\pi}{2}} \left[-\frac{1}{3}\sqrt{(a^2-r^2)^3}\right]_0^b d\theta = 8\int_0^{\frac{\pi}{2}} -\frac{1}{3}\left(\sqrt{(a^2-b^2)^3} - a^3\right) d\theta8∫02π∫0ba2−r2⋅rdrdθ=8∫02π[−31(a2−r2)3]0bdθ=8∫02π−31((a2−b2)3−a3)dθ =8⋅π2⋅13(a3−(a2−b2)3)=43π{a3−(a2−b2)3}= 8 \cdot \frac{\pi}{2} \cdot \frac{1}{3}\left(a^3 - \sqrt{(a^2-b^2)^3}\right) = \frac{4}{3}\pi\left\{a^3 - \sqrt{(a^2-b^2)^3}\right\}=8⋅2π⋅31(a3−(a2−b2)3)=34π{a3−(a2−b2)3}