Answer
$$\frac{{45}}{{32}}\left( {12 + {\pi ^2}} \right)$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\int_0^{1 + \sin \theta } {15\theta r} dr} d\theta \cr
& {\text{Let }}\theta = x{\text{ and }}r = y \cr
& \int_0^{\pi /2} {\int_0^{1 + \sin \theta } {15\theta r} dr} d\theta = \int_0^{\pi /2} {\int_0^{1 + \sin x} {15xy} dy} dx \cr
& {\text{Using a computer algebra system to evaluate the }} \cr
& {\text{iterated integral, we obtain:}}{\text{}} \cr
& \int_0^{\pi /2} {\int_0^{1 + \sin \theta } {15\theta r} dr} d\theta = \frac{{45}}{{32}}\left( {12 + {\pi ^2}} \right) \approx 30.7541 \cr} $$
