Answer
$$\displaystyle{V=\frac{26\pi }{3}}\\$$
Work Step by Step
$\displaystyle{A\left(x\right)=\pi \left(x+1\right)^2}\\ \displaystyle{A\left(x\right)=\pi \left(x^2+2x+1\right)}\\\\$
$\displaystyle{V=\int_0^2A\left(x\right)\ dx}\\ \displaystyle{V=\int_0^2\pi \left(x^2+2x+1\right)\ dx}\\ \displaystyle{V=\pi \int_0^2\left(x^2+2x+1\right)\ dx}\\ \displaystyle{V=\pi\left[\frac{1}{3}x^3+ x^2+ x\right]_0^2}\\ \displaystyle{V=\pi\left(\left(\frac{1}{3}2^3+ 2^2+ 2\right)-\left(0\right)\right)}\\ \displaystyle{V=\frac{26\pi }{3}}\\ $