Answer
$A = \frac{{4\pi }}{3} + \ln \left| {\frac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }}} \right|$
Work Step by Step
$$\eqalign{
& y = \csc x,{\text{ }}y = 2,{\text{ }}\frac{\pi }{6} \leqslant x \leqslant \frac{{5\pi }}{6} \cr
& 2 \geqslant \csc x{\text{ on the interval }}\left( {\frac{\pi }{6},\frac{{5\pi }}{6}} \right) \cr
& {\text{The area is given by}} \cr
& A = \int_{\pi /6}^{5\pi /6} {\left( {2 - \csc x} \right)} dx \cr
& {\text{Integrate and evaluate}} \cr
& A = \left[ {2x - \ln \left| {\csc x - \cot x} \right|} \right]_{\pi /6}^{5\pi /6} \cr
& A = \left[ {2\left( {\frac{{5\pi }}{6}} \right) - \ln \left| {\csc \left( {\frac{{5\pi }}{6}} \right) - \cot \left( {\frac{{5\pi }}{6}} \right)} \right|} \right] \cr
& {\text{ }} - \left[ {2\left( {\frac{\pi }{6}} \right) - \ln \left| {\csc \left( {\frac{\pi }{6}} \right) - \cot \left( {\frac{\pi }{6}} \right)} \right|} \right] \cr
& A = \left[ {\frac{{5\pi }}{3} - \ln \left| {2 + \sqrt 3 } \right|} \right] - \left[ {\frac{\pi }{3} - \ln \left| {2 - \sqrt 3 } \right|} \right] \cr
& A = \frac{{5\pi }}{3} - \ln \left| {2 + \sqrt 3 } \right| - \frac{\pi }{3} + \ln \left| {2 - \sqrt 3 } \right| \cr
& A = \frac{{4\pi }}{3} + \ln \left| {\frac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }}} \right| \cr
& A \approx 1.5549 \cr} $$
