6.2 Volumes/1

Failed to parse (unknown function "\x"): {\displaystyle \begin{align} y=2-\frac{1}{2}x, \x-axis\\[1ex] y=0\\[1ex] x=1\\[1ex] x=2\\[1ex] \end{align} }

${\begin{aligned}\pi \int _{1}^{2}\left[(2-{\frac {1}{2}}x)^{2}\right]dy&=\pi \int _{1}^{2}\left[(4-2x+{\frac {1}{4}}x^{2})\right]dx\\[2ex]&=\pi \left[4x-x^{2}+{\frac {1}{12}}x^{3}\right]{\Bigg |}_{1}^{2}\\[2ex]&=\pi \left[4(2)-(2)^{2}+{\frac {1}{12}}(2)^{3}-(4(1)-(1)^{2}+{\frac {1}{12}}(1)^{3})\right]=\pi \left[8-4+{\frac {8}{12}}-(4-1+{\frac {1}{2}})\right]\\[2ex]&=\pi \left[4+{\frac {8}{12}}-3-{\frac {1}{3}}\right]=\pi \left[1+{\frac {7}{12}}\right]\\[2ex]&=\pi \left[{\frac {12}{12}}+{\frac {7}{12}}\right]=\pi \left[{\frac {19}{12}}\right]\\[2ex]&={\frac {19\pi }{12}}\end{aligned}}$