Mathematics - Area Under Curves Question with Solution | TestHub

$\begin{array}{rl}& {\mathrm{x}}^2+{\mathrm{y}}^2=5\\ & {\mathrm{x}}^2+(\mathrm{x}-1{)}^2=25\Rightarrow \mathrm{x}=4\\ & {\mathrm{x}}^2+(-\mathrm{x}+1{)}^2=5\Rightarrow \mathrm{x}=-3\\ & \mathrm{A}=25\pi -{\int }_{-3}^4\sqrt{25-{\mathrm{x}}^2}\mathrm{dx}+\frac{1}{2}\times 4\times 4+\frac{1}{2}\times 3\times 3\\ & \mathrm{A}=25\pi +\frac{25}{2}-{\left[\frac{\mathrm{x}}{2}\sqrt{25-{\mathrm{x}}^2}+\frac{25}{2}{\sin }^{-1}\frac{\mathrm{x}}{5}\right]}_{-3}^4\\ & \mathrm{A}=25\pi +\frac{25}{2}-\left[6+\frac{25}{2}{\sin }^{-1}\frac{4}{5}+6+\frac{25}{2}{\sin }^{-1}\frac{3}{5}\right]\\ & \mathrm{A}=25\pi +\frac{1}{2}-\frac{25}{2}\cdot \frac{\pi }{2}\\ & \mathrm{A}=\frac{75\pi }{4}+\frac{1}{2}\\ & \mathrm{A}=\frac{1}{4}(75\pi +2)\\ & \mathrm{b}=75,\mathrm{c}=2\\ & \mathrm{b}+\mathrm{c}=75+2=77\end{array}$