Chapter 5 - Applications of Integration - 5.3 Volumes by Cylindrical Shells - 5.3 Exercises - Page 382: 16

$\displaystyle{V=\frac{40\pi}{3}}$

$\displaystyle{y=4-2x}\\ \displaystyle{0=4-2x}\\ \displaystyle{2x=4}\\ \displaystyle{x=2}$ $\displaystyle{V=\int_{0}^{2}(2\pi (x+1))\left(4-2x\right)\ dx}\\ \displaystyle{V=2\pi\int_{0}^{2}2x-2x^2+4\ dx}\\ \displaystyle{V=2\pi\left[x^2-\frac{2}{3}x^3+4x\right]_{0}^{2}}\\ \displaystyle{V=2\pi\left(\left((2)^2-\frac{2}{3}(2)^3+4(2)\right)-(0)\right)}\\ \displaystyle{V=\frac{40\pi}{3}}$