6.2 Volumes/3

$y={\frac {1}{x}},x=1,x=2,y=0;{\text{about the x-axis}}$

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