Hi I just need a check on if the answer I got is correct.

"Find the volume of the solid obtained by rotating the region between $\displaystyle y = \frac {1}{x}$, with 1 $\displaystyle \leq $ x $\displaystyle \leq 2$, about the y-axis"

y(1) = 1

y(2) = $\displaystyle \frac{1}{2}$

x = $\displaystyle \frac{1}{y}$

$\displaystyle \pi \int$ (from 1 to $\displaystyle \frac {1}{2}$) ,$\displaystyle \frac{1}{y^2}$ dy + $\displaystyle \int$(from $\displaystyle \frac{1}{2}$ to 0) $\displaystyle 2^2$ dy

= $\displaystyle \pi [-\frac{1}{y}]\frac{1}{\frac{1}{2}}$ + $\displaystyle 2^2[y] \frac{\frac{1}{2}}{0}$

= $\displaystyle \pi$ [ -1 + 2 + 2]

= $\displaystyle 3\pi$

Any help would be great thanks.