Chapter 6 - Section 6.3 - Volumes by Cylindrical Shells - 6.3 Exercises - Page 454: 10

$\displaystyle{V=8\pi}\\$

$\displaystyle{V=\int_{0}^{2}(2\pi y)\left(y^2\right)\ dy}\\ \displaystyle{V=2\pi\int_{0}^{2}y^3\ dy}\\ \displaystyle{V=2\pi\left[\frac{1}{4}y^4\right]_{0}^{2}}\\ \displaystyle{V=2\pi\left(\left(\frac{1}{4}(2)^4\right)-(0)\right)}\\ \displaystyle{V=8\pi}\\$