6.2 Volumes/1

${\begin{aligned}y=2-{\frac {1}{2}}x,x-axis\\[1ex]y=0\\[1ex]x=1\\[1ex]x=2\\\end{aligned}}$

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