Answer
$\pi$
Work Step by Step
$V$ = $\int_1^{e}2\pi{x}·[\ln{x}-(\ln{x})^{2}]dx$ = $2\pi{\int_1^{e}x\ln{x}dx}$ - $2\pi{\int_1^{e}x(\ln{x})^{2}dx}$
use integration by parts
$u$ = $(\ln{x})^{2}$
$u'$ = $\frac{2\ln{x}}{x}$
$v'$ = $x$
$v$ = $\frac{x^{2}}{2}$
$V$ = $2\pi{\int_1^{e}x\ln{x}dx}$ - $2\pi(\frac{1}{2}x^{2})(\ln{x})^{2}|_1^e$ - $\int_1^{e}x\ln{x}dx$ = $-\pi{e^{2}}+4\pi{\int_1^{e}x\ln{x}dx}$
use integration by parts again
$u$ = $\ln{x}$
$u'$ = $\frac{1}{x}$
$v'$ = $x$
$v$ = $\frac{x^{2}}{2}$
$V$ = $-\pi{e^{2}}+4\pi[(\frac{1}{2}x^{2})\ln{x}|_1^e-\frac{1}{2}\int_1^e{{xdx}}]$ = $\pi$
