Answer
$\displaystyle{V=2\pi\left(\frac{4\pi }{3}-\sqrt{3}\right)}\\ $
Work Step by Step
$\displaystyle{3=1+\sec x}\\
\displaystyle{\sec x=2}\\
\displaystyle{\cos x=\frac{1}{2}}\\
\displaystyle{x=-\frac{\pi}{3} \qquad x=\frac{\pi}{3}}$
$\displaystyle{A\left(x\right)=\pi \left(1-3\right)^2-\pi \left(1-\left(1+\sec x\right)\right)^2}\\
\displaystyle{A\left(x\right)=\pi \left(4-\sec^2x\right)}\\$
$\displaystyle{V=\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}A\left(x\right)\ dx}\\
\displaystyle{V=\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\pi \left(4-\sec^2x\right)\ dx}\\
\displaystyle{V=\pi \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}4-\sec^2x\ dx}\\$
By symmetry
$\displaystyle{V=2\pi \int_{0}^{\frac{\pi}{3}}4-\sec^2x\ dx}\\
\displaystyle{V=2\pi\left[4x-\tan x\right]_{0}^{\frac{\pi}{3}}}\\
\displaystyle{V=2\pi\left(\left(4\times\left({\frac{\pi}{3}}\right)-\tan \left({\frac{\pi}{3}}\right)\right)-\left(4(0)-\tan (0)\right)\right)}\\
\displaystyle{V=2\pi\left(\frac{4\pi }{3}-\sqrt{3}\right)}\\ $
