6.2 Volumes/25

$R+f(y)=1R=1-f(y)r=1{\begin{aligned}2x^{2}+3(x-1)(x-2)&=2x^{2}+3(x^{2}-3x+2)\\&=2x^{2}+3x^{2}-9x+6\\&=5x^{2}-9x+6\end{aligned}}$

${\begin{aligned}\pi \int _{0}^{1}\left[(1)^{2}-(1-y^{2})^{2}\right]dy&=\pi \int _{0}^{1}\left[(1-(1-2y^{2}+y^{4})\right]dy=\pi \int _{0}^{1}\left[(2y^{2}-y^{4})\right]dy\\[2ex]&=\pi \left[{\frac {2y^{3}}{3}}-{\frac {y^{5}}{5}}\right]{\Bigg |}_{0}^{1}\\[2ex]&=\pi \left[{\frac {2}{3}}-{\frac {1}{5}}\right]=\pi \left[{\frac {10}{15}}-{\frac {3}{15}}\right]\\[2ex]&={\frac {7\pi }{15}}\end{aligned}}$