6.2 Volumes/25

Failed to parse (unknown function "\nonumber"): {\displaystyle R + f(y) = 1 R = 1-f(y) r = 1 \begin{align} 2x^2 + 3(x-1)(x-2) & = 2x^2 + 3(x^2-3x+2)\\ \nonumber &= 2x^2 + 3x^2 - 9x + 6\\ &= 5x^2 - 9x + 6 \end{align} }

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