6.2 Volumes/1

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

${\begin{aligned}\pi \int _{1}^{2}\left[(2-{\frac {1}{2}}x)^{2}\right]dy&=\pi \int _{1}^{2}\left[(4-2x+{\frac {1}{4}}x^{2})\right]dx\\[2ex]&=\pi \left[4x-x^{2}+{\frac {1}{12}}x^{3}\right]{\Bigg |}_{1}^{2}\\[2ex]&=\pi \left[4(2)-(2)^{2}+{\frac {1}{12}}(2)^{3}-(4(1)-(1)^{2}+{\frac {1}{12}}(1)^{3})\right]=\pi \left[8-4+{\frac {8}{12}}-(4-1+{\frac {1}{2}})\right]\\[2ex]&={\frac {7\pi }{15}}\end{aligned}}$