활꼴 회전체 부피 차

$\displaystyle \begin{aligned} \mathrm{V} _{1} & = \pi \int _{\frac{1}{2}} ^{1} {} x ^{2} dy = \pi \int _{\frac{1}{2}} ^{1} {} \left( 1 - y ^{2} \right) dy \end{aligned}$$\displaystyle = \pi \left[ y - \frac{1}{3} y ^{3} \right] _{\frac{1}{2}} ^{1} = \pi \left\{ \left( 1 - \frac{1}{2} \right) - \frac{1}{3} \left( 1 ^{3} - \left( \frac{1}{2} \right) ^{3} \right) \right\} = \frac{5}{24} \pi$ $\displaystyle \begin{aligned} \mathrm{V} _{2} & = \pi \int _{- \frac{1}{2}} ^{\frac{1}{2}} {} x ^{2} dy - \pi \left( \frac{\sqrt{3}}{2} \right) ^{2} \end{aligned}$ $\displaystyle = \pi \int _{- \frac{1}{2}} ^{\frac{1}{2}} {} \left( 1 - y ^{2} \right) dy - \pi \left( \frac{\sqrt{3}}{2} \right) ^{2}$ $\displaystyle = 2 \pi \int _{0} ^{\frac{1}{2}} {} \left( 1 - y ^{2} \right) dy - \pi \left( \frac{\sqrt{3}}{2} \right) ^{2}$ $\displaystyle \begin{aligned} & = 2 \pi \left[ y - \frac{1}{3} y ^{3} \right] _{0} ^{\frac{1}{2}} - \pi \left( \frac{\sqrt{3}}{2} \right) ^{2} \end{aligned}$ $\displaystyle = 2 \pi \left\{ \left( \frac{1}{2} - 0 \right) - \frac{1}{3} \left( \left( \frac{1}{2} \right) ^{3} - 0 ^{3} \right) \right\} - \pi \left( \frac{\sqrt{3}}{2} \right) ^{2} = \frac{4}{24} \pi$ $\displaystyle \therefore \mathrm{V} _{1} - \mathrm{V} _{2} = \frac{5}{24} \pi - \frac{4}{24} \pi = \frac{\pi}{24}$