Answer
$\frac{3\pi}{5}$
Work Step by Step
Step 1. Draw a diagram as shown in the figure. The limits of integration with respect to $x$ can be found as from $-1$ to $0$.
Step 2. Approximate using cylinders; we have
$dV=2\pi (x+2)(-x^3-0)) dx=-2\pi (x^4+2x^3)\ dx$
Step 3. We have
$V=-2\pi\int_{-1}^0 (x^4+2x^3)\ dx=-2\pi (\frac{1}{5}x^5+\frac{1}{2}x^4)|_{-1}^0 =2\pi (\frac{1}{5}(-1)^5+\frac{1}{2}(-1)^4)=\frac{3\pi}{5}$
