Chapter 7: Transcendental Functions - Section 7.2 - Natural Logarithms - Exercises 7.2 - Page 382: 75

$\pi\ln{16}$

$V$ = $\int_{{\,a}}^{{\,b}} A(x)$ $dx$ $V$ = $\int_{{\,\frac{1}{2}}}^{{\,2}}2\pi(x)(\frac{1}{x^{2}})$ $dx$ $V$ = $\int_{{\,\frac{1}{2}}}^{{\,2}}2\pi(\frac{1}{x})$ $dx$ $V$ = $2\pi(\ln{x})$$|_{{\,\frac{1}{2}}}^{{\,2}}$ $V$ = $2\pi(\ln{2}-\ln{\frac{1}{2}})$ $V$ = $4\pi\ln{2}$ = $\pi\ln{16}$