Question

$$\begin{matrix} & A = \int_{0}^{2}\mspace{2mu}\mspace{2mu} 2\sqrt{x}dx + 2\int_{1}^{\sqrt{2}}\mspace{2mu}\mspace{2mu}\sqrt{8 - 4x^{2}}dx \\ & \ = \frac{8}{3}\left( x^{\frac{3}{2}} \right)\int_{0}^{1}\mspace{2mu}\mspace{2mu} + 4\int_{1}^{\sqrt{2}}\mspace{2mu}\mspace{2mu}\sqrt{2 - x^{2}}dx \\ & \ = \frac{8}{3} + 4 \times \left. \ \frac{1}{2}\left\lbrack x\sqrt{2 - x^{2}} + 2\sin^{- 1}\left( \frac{x}{\sqrt{2}} \right) \right\rbrack \right|_{1}^{\sqrt{2}} \\ & \ = \frac{8}{3} + 2\left\lbrack 2 \times \frac{\pi}{2} - 1 - 2 \times \frac{\pi}{4} \right\rbrack \\ & \ = \frac{8}{3} + 2\pi - 2 - \pi = \pi + \frac{2}{3}\text{~}\text{sq. units}\text{~} \end{matrix}$$