Mathematics-I (3110014)

${\int }_0^{\mathrm{\pi }}{\int }_0^{\sin \left(\mathrm{\theta }\right)}\mathrm{r}d\mathrm{r} d\mathrm{\theta }$

$={\int }_0^{\mathrm{\pi }}{\left[\frac{{\mathrm{r}}^2}{2}\right]}_0^{\sin \left(\mathrm{\theta }\right)} d\mathrm{\theta }$

$={\int }_0^{\mathrm{\pi }}\left[\frac{{\sin }^2\left(\mathrm{\theta }\right)-0}{2}\right] d\mathrm{\theta }$

$={\int }_0^{\mathrm{\pi }}\left[\frac{1-\cos \left(2\mathrm{\theta }\right)}{4}\right] d\mathrm{\theta }$

$=\frac{1}{4}{\int }_0^{\mathrm{\pi }}\left[1-\cos \left(2\mathrm{\theta }\right)\right] d\mathrm{\theta }$

$=\frac{1}{4}{\left[\mathrm{\theta }-\frac{\sin \left(2\mathrm{\theta }\right)}{2}\right]}_0^{\mathrm{\pi }}$

$=\frac{1}{4}\left[\left\{\mathrm{\pi }-\frac{\sin \left(2\mathrm{\pi }\right)}{2}\right\}-\left\{0-\frac{\sin \left(0\right)}{2}\right\}\right]$

$=\frac{1}{4}\left[\left\{\mathrm{\pi }-0\right\}-\left\{0-0\right\}\right]$

$=\frac{\mathrm{\pi }}{4}$

$\mathrm{So}, {\int }_0^{\mathrm{\pi }}{\int }_0^{\sin \left(\mathrm{\theta }\right)}\mathrm{r}d\mathrm{r} d\mathrm{\theta }=\frac{\mathrm{\pi }}{4}$