Chapter 6 - Section 6.3 - Volumes by Cylindrical Shells - 6.3 Exercises - Page 454: 9

$\displaystyle{V=4\pi}\\$

$\displaystyle{V=\int_{1}^{3}(2\pi y)\left(\frac{1}{y}\right)\ dy}\\ \displaystyle{V=2\pi\int_{1}^{3}1\ dy}\\ \displaystyle{V=2\pi\left[y\right]_{1}^{3}}\\ \displaystyle{V=2\pi\left(\left(3\right)-(1)\right)}\\ \displaystyle{V=4\pi}\\$