Answer
$$\frac{3}{4}\pi $$
Work Step by Step
$$\eqalign{
& \int_0^\pi {\int_0^{1 + \cos \theta } r } drd\theta \cr
& = \int_0^\pi {\left[ {\int_0^{1 + \cos \theta } r dr} \right]} d\theta \cr
& {\text{solve the inner integral and treat }}\theta {\text{ as a constant}} \cr
& = \int_0^{1 + \cos \theta } r dr \cr
& {\text{then}} \cr
& = \left[ {\frac{{{r^2}}}{2}} \right]_0^{1 + \cos \theta } \cr
& {\text{evaluating the limits in the variable }}r \cr
& = \frac{{{{\left( {1 + \cos \theta } \right)}^2}}}{2} - \frac{{{{\left( 0 \right)}^2}}}{2} \cr
& = \frac{{1 + 2\cos \theta + {{\cos }^2}\theta }}{2} \cr
& {\text{simplifying}} \cr
& = \frac{1}{2} + \cos \theta + \frac{1}{2}{\cos ^2}\theta \cr
& {\text{then}} \cr
& \int_0^\pi {\left[ {\int_0^{1 + \cos \theta } r dr} \right]} d\theta = \int_0^\pi {\left( {\frac{1}{2} + \cos \theta + \frac{1}{2}{{\cos }^2}\theta } \right)} d\theta \cr
& = \int_0^\pi {\left( {\frac{1}{2} + \cos \theta + \frac{1}{4}\left( {1 + \cos 2\theta } \right)} \right)} d\theta \cr
& = \int_0^\pi {\left( {\frac{3}{4} + \cos \theta + \frac{1}{4}\cos 2\theta } \right)} d\theta \cr
& {\text{integrating }} \cr
& = \left( {\frac{3}{4}\theta + \sin \theta + \frac{1}{8}\sin 2\theta } \right)_0^\pi \cr
& {\text{evaluate}} \cr
& = \left( {\frac{3}{4}\pi + \sin \pi + \frac{1}{8}\sin 2\pi } \right) - \left( {\frac{3}{4}\left( 0 \right) + \sin \left( 0 \right) + \frac{1}{8}\sin 2\left( 0 \right)} \right) \cr
& = \frac{3}{4}\pi \cr} $$
