Answer
$\displaystyle{V=\frac{3\pi }{4}}\\ $
Work Step by Step
$\displaystyle{A\left(x\right)=\pi \left(\frac{1}{x}\right)^2}\\ \displaystyle{A\left(x\right)=\pi \left(\frac{1}{x^2}\right)}\\
\displaystyle{A\left(x\right)=\pi x^{-2}}\\$
$\displaystyle{V=\int_1^4A\left(x\right)\ dx}\\
\displaystyle{V=\int_1^4\pi x^{-2}\ dx}\\
\displaystyle{V=\pi \int_1^4 x^{-2}\ dx}\\
\displaystyle{V=\pi\left[-x^{-1}\right]_1^4}\\
\displaystyle{V=\pi\left(\left(-4^{-1}\right)-\left(-1^{-1}\right)\right)}\\
\displaystyle{V=\frac{3\pi }{4}}\\ $
