회전체의 부피

$x$축으로 회전시킨 입체의 부피는 $\displaystyle \pi \int _{a} ^{b} {} y ^{2} dx$이므로 $\displaystyle V = \int _{0} ^{1} {} \left\{ x ( 1 - x ) \right\} ^{2} dx$$\displaystyle = \int _{0} ^{1} {} ( x - x ^{2} ) ^{2} dx$ $\displaystyle = \int _{0} ^{1} {} ( x ^{4} - 2 x ^{3} + x ^{2} ) dx$ $\displaystyle = \pi \left[ \frac{1}{5} x ^{5} - \frac{1}{2} x ^{4} + \frac{1}{3} x ^{3} \right] _{0} ^{1}$ $\displaystyle = \pi \left( \frac{1}{5} - \frac{1}{2} + \frac{1}{3} \right)$$\displaystyle = \frac{1}{30} \pi$