Chapter 14 - Multiple Integrals - 14.3 Double Integrals In Polar Coordinates - Exercises Set 14.3 - Page 1024: 3

$$\frac{{2{a^3}}}{9}$$

$$\eqalign{ & \int_0^{\pi /2} {\int_0^{a\sin \theta } {{r^2}} } drd\theta \cr & = \int_0^{\pi /2} {\left[ {\int_0^{a\sin \theta } {{r^2}} dr} \right]} d\theta \cr & {\text{solve the inner integral and treat }}\theta {\text{ as a constant}} \cr & = \int_0^{a\sin \theta } {{r^2}} dr \cr & {\text{then}} \cr & = \left[ {\frac{{{r^3}}}{3}} \right]_0^{a\sin \theta } \cr & {\text{evaluating the limits in the variable }}r \cr & = \frac{1}{3}\left( {{{\left( {a\sin \theta } \right)}^3} - {{\left( 0 \right)}^3}} \right) \cr & = \frac{1}{3}\left( {{a^3}{{\sin }^3}\theta - 0} \right) \cr & {\text{simplifying}} \cr & = \frac{{{a^3}{{\sin }^3}\theta }}{3} \cr & {\text{then}} \cr & \int_0^{\pi /2} {\left[ {\int_0^{a\sin \theta } {{r^2}} dr} \right]} d\theta = \int_0^{\pi /2} {\frac{{{a^3}{{\sin }^3}\theta }}{3}} d\theta \cr & = \frac{{{a^3}}}{3}\int_0^{\pi /2} {{{\sin }^2}\theta \sin \theta } d\theta \cr & = \frac{{{a^3}}}{3}\int_0^{\pi /2} {\left( {1 - {{\cos }^2}\theta } \right)\sin \theta } d\theta \cr & = \frac{{{a^3}}}{3}\int_0^{\pi /2} {\left( {\sin \theta + {{\cos }^2}\theta \left( { - \sin \theta } \right)} \right)} d\theta \cr & {\text{integrating }} \cr & = \frac{{{a^3}}}{3}\left( { - \cos \theta + \frac{{{{\cos }^3}\theta }}{3}} \right)_0^{\pi /2} \cr & {\text{evaluate}} \cr & = \frac{{{a^3}}}{3}\left( { - \cos \left( {\frac{\pi }{2}} \right) + \frac{{{{\cos }^3}\left( {\pi /2} \right)}}{3}} \right) - \frac{{{a^3}}}{3}\left( { - \cos \left( 0 \right) + \frac{{{{\cos }^3}\left( 0 \right)}}{3}} \right) \cr & = \frac{{{a^3}}}{3}\left( {0 + \frac{0}{3}} \right) - \frac{{{a^3}}}{3}\left( { - 1 + \frac{1}{3}} \right) \cr & = \frac{{{a^3}}}{3}\left( 0 \right) - \frac{{{a^3}}}{3}\left( { - \frac{2}{3}} \right) \cr & = \frac{{2{a^3}}}{9} \cr} $$