A) \[24\,\,{{\sin }^{-1}}\left( \frac{4}{5} \right)\]
B) \[25\,\,{{\sin }^{-1}}\left( \frac{4}{5} \right)\]
C) \[2+\frac{25}{2}{{\sin }^{-1}}\left( \frac{4}{5} \right)\]
D) None of these
Correct Answer: C
Solution :
Required area \[=\int_{0}^{4}{\sqrt{(25-{{x}^{2}})}}dx-\int_{0}^{2}{\frac{4-{{x}^{2}}}{4}}dx-\int_{2}^{4}{\frac{{{x}^{2}}-4}{4}}dx\] \[=\left[ \frac{x}{2}\sqrt{25-{{x}^{2}}}+\frac{25}{2}\,{{\sin }^{-1}}\frac{x}{5} \right]_{0}^{4}\] \[-\frac{1}{4}\left[ 4x-\frac{{{x}^{3}}}{3} \right]_{0}^{2}-\frac{1}{4}\left[ \frac{{{x}^{3}}}{3}-4x \right]_{2}^{4}\] \[=\left( 2\times 3+\frac{25}{2}\,{{\sin }^{-1}}\left( \frac{4}{5} \right) \right)-\frac{1}{4}\left( 8-\frac{8}{3} \right)\] \[-\frac{1}{4}\left( \frac{64}{3}-16 \right)+\frac{1}{4}\left( \frac{8}{3}-8 \right)\] \[=2+\frac{25}{2}si{{n}^{-1}}\left( \frac{4}{5} \right)\]You need to login to perform this action.
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