Answer
$\pi\ln2$
Work Step by Step
$V$ = $\int_{{\,a}}^{{\,b}} A(x)$ $dx$
$V$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{2}}}\pi(\sqrt{cot(x)})^{2}$ $dx$
$V$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{2}}}\pi{cot(x)}$ $dx$
$V$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{2}}}\pi{\frac{cos(x)}{sin(x)}}$ $dx$
$u$ = $sin(x)$
$du$ = $cos(x)$$dx$
so
$V$ = $\pi\int{\frac{1}{u}}$ $du$
$V$ = $\pi\ln{u}$
$V$ = $\pi{\ln(sin(x))}$$|_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{2}}}$
$V$ = $\pi[{\ln(sin(\frac{\pi}{2}))}-{\ln(sin(\frac{\pi}{6})}]$
$V$ = $\pi\ln2$
