Mathematics - Area Under Curves Question with Solution | TestHub

Required area $=\mathrm{Ar}($ circle from 0 to 2$)-$ $\mathrm{ar}($ para from 0 to 2$)$ $\begin{array}{rl}& ={\int }_0^2\sqrt{8-x^2}dx-{\int }_0^2\sqrt{2x}dx\\ & ={\left[\frac{x}{2}\sqrt{8-x^2}+\frac{8}{2}{\sin }^{-1}\frac{x}{2\sqrt{2}}\right]}_0^2-\sqrt{2}{\left[\frac{x\sqrt{x}}{3/2}\right]}_0^2\\ & =\frac{2}{2}\sqrt{8-4}+\frac{8}{2}{\sin }^{-1}\frac{2}{2\sqrt{2}}-\frac{2\sqrt{2}}{3}(2\sqrt{2}-0)\\ & \Rightarrow 2+4\cdot \frac{\pi }{4}-\frac{8}{3}=\pi -\frac{2}{3}\end{array}$