6.2 Volumes/15

$x=y^{2},x=1;{\text{about x=1}}$

${\begin{aligned}\pi \int _{-1}^{1}\left(1-y^{2}\right)^{2}dy&=\pi \int _{-1}^{1}\left(1-2y^{2}+y^{4}\right)dy\\[2ex]&=\pi \left[y-{\frac {2}{3}}y^{3}+{\frac {y^{5}}{5}}\right]{\Bigg |}_{-1}^{1}\\[2ex]&=\pi \left[1+{\frac {2}{3}}+{\frac {1}{5}}\right]-\pi \left[-1+{\frac {2}{5}}-{\frac {1}{5}}\right]\\[2ex]&={\frac {16\pi }{15}}\end{aligned}}$