y = 1 x , x = 1 , x = 2 , y = 0 ; about the x-axis {\displaystyle y={\frac {1}{x}},x=1,x=2,y=0;{\text{about the x-axis}}}
π ∫ 1 2 [ ( 1 x ) 2 ] d x = π ∫ 1 2 [ ( 1 x 2 ) ] d x = π [ − 1 x ] | 1 2 = π [ ( − 1 ( 2 ) ) − ( − 1 ( 1 ) ) ] = π [ − 1 2 + 1 ] = π [ 1 2 ] = π 2 {\displaystyle {\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}}}
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/329b1685d411fbea079953fc7dd6d29610910a79)