Question

answer this question
​

Answer 1

Step-by-step explanation:

We have,

$\int \limits_{0}^{ \frac{\pi}{2} } \int \limits_{0}^{ \cos( \theta) } {r}^{2} drd \theta$

$\int \limits_{0}^{ \frac{\pi}{2} } [ \frac{ {r}^{3} }{3} ]^{ \cos(\theta) } _{0} d \theta$

$\int \limits_{0}^{ \frac{\pi}{2} } \frac{ { \cos}^{3}( \theta) }{3} d \theta$

$= \frac{1}{3} \int \limits_{0}^{ \frac{\pi}{2} } (1 - \sin^{2} ( \theta)) \cos( \theta) d \theta$

$Let \:sin(\theta)=t$

$\implies\cos(\theta)d\theta=dt$

$= \frac{1}{3} \int \limits_{0}^{ 1 } (1 - {t}^{2} ) dt$

$= \frac{1}{3} [t - \frac{ {t}^{3} }{3} ]^{1}_{0}$

$= \frac{1}{3} (1 - \frac{1}{3} )$

$= \frac{1}{3} \times \frac{2}{3}$

$= \frac{2}{9}$