Answer
$$A = \frac{8}{{3\ln 3}}$$
Work Step by Step
$$\eqalign{
& y = {3^{\cos x}}\sin x,{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = \pi \cr
& {\text{From the graph, we can define the area enclosed by}} \cr
& A = \int_0^\pi {{3^{\cos x}}\sin x} dx \cr
& {\text{Integrating}} \cr
& A = - \frac{1}{{\ln 3}}\left[ {{3^{\cos x}}} \right]_0^\pi \cr
& A = - \frac{1}{{\ln 3}}\left[ {{3^{\cos \pi }} - {3^{\cos 0}}} \right] \cr
& A = - \frac{1}{{\ln 3}}\left[ {{3^{ - 1}} - 3} \right] \cr
& A = \frac{8}{{3\ln 3}} \approx 2.427 \cr} $$