Explanation

(c)
$Required\ area = 2\left\lbrack \int_{0}^{4}{\sqrt{(25) - x^{2})}dx - \int_{0}^{2}{\frac{4 - x^{2}}{4}dx - \int_{2}^{4}{\frac{x^{2} - 4}{4}dx}}} \right\rbrack$
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$= 2\left\lbrack \left\lbrack \frac{x}{2}\sqrt{25 - x^{2}} + \frac{25}{2}\sin^{- 1}\frac{x}{5}\ \right\rbrack_{0}^{4} - \frac{1}{4}\left\lbrack 4x - \frac{x^{3}}{3} \right\rbrack_{0}^{2} - \frac{1}{4}\left\lbrack \frac{x^{3}}{3} - 4x \right\rbrack_{2}^{4} \right\rbrack_{\ }^{\ }$
$= 2\left\lbrack \left\lbrack 2 \times 3 + \frac{25}{5}\sin^{- 1}\left( \frac{4}{5} \right) - \frac{1}{4}\left( \frac{64}{3} - 16 \right) + \frac{1}{4}\left( \frac{8}{3} - 8 \right) \right\rbrack \right\rbrack = 4 + \sin^{- 1}\left( \frac{4}{5} \right)$