Answer
$$\frac{{15\pi }}{2}$$
Work Step by Step
$$\eqalign{
& \int_0^{2\pi } {\int_0^{1 + \cos \theta } {6{r^2}\cos \theta } dr} d\theta \cr
& {\text{Let }}r = x{\text{ and }}\theta = y \cr
& \int_0^{2\pi } {\int_0^{1 + \cos \theta } {6{r^2}\cos \theta } dr} d\theta = \int_0^{2\pi } {\int_0^{1 + \cos y} {6{x^2}\cos y} dx} dy \cr
& {\text{Using a computer algebra system to evaluate the }} \cr
& {\text{iterated integral, we obtain:}}{\text{}} \cr
& \int_0^{2\pi } {\int_0^{1 + \cos \theta } {6{r^2}\cos \theta } dr} d\theta = \frac{{15\pi }}{2} \approx 23.5619 \cr} $$
