旋轉圖形建立座標系如下圖：
陰影部份上下曲邊公式如下：
\begin{cases} x^2+y^2=5^2 \\ x^2+(y+5\sqrt{2})^2=10^2 \end{cases}
求解交點座標：
(y+5\sqrt{2})^2-y^2=75 \to 10\sqrt{2}y+50=75
\begin{cases} y=\frac{5\sqrt{2}}{4} \\ x=\sqrt{\frac{175}{8}} \end{cases}
用積分法求解陰影面積：
\begin{align} \frac{S}{4} & =\int_0^{\sqrt{\frac{175}{8}}}\sqrt{(5^2-x^2)}-(\sqrt{(10^2-x^2)}-5\sqrt{2})\mathrm{d}x \\ & = \int_0^{\sqrt{\frac{175}{8}}}\sqrt{(5^2-x^2)}\mathrm{d}x - \int_0^{\sqrt{\frac{175}{8}}}\sqrt{(10^2-x^2)}\mathrm{d}x + 5\sqrt{2} \cdot \sqrt{\frac{175}{8}} \end{align}
查常用積分表，可得：
\int \sqrt{a^2 - x^2}\mathrm{d}{x} = \frac12 \left(x\sqrt{a^2 - x^2} + a^2\arcsin\frac xa\right) + C
\begin{align} \frac{S}{4} & =\left[\frac{25}{16}\sqrt{7} + \frac{25}{2}\arcsin(\frac{\sqrt{\frac{7}{2}}}{2})\right] - \left[\frac{125}{16}\sqrt{7} + 50\arcsin(\frac{\sqrt{\frac{7}{2}}}{4})\right] + \frac{25}{2}\sqrt{7} \\ & = \frac{25}{4}\sqrt{7} + \frac{25}{2}\arcsin(\frac{\sqrt{\frac{7}{2}}}{2}) - 50\arcsin(\frac{\sqrt{\frac{7}{2}}}{4}) \end{align}
S=25\sqrt{7} + 50\arcsin(\frac{\sqrt{\frac{7}{2}}}{2}) - 200\arcsin(\frac{\sqrt{\frac{7}{2}}}{4})\approx 29.27625
