Let [] be the area of the region [ (x,y) R^2:x^2 + y^2 21,y^2 4x,x 1 ].

Question

Let $$\Delta$$ be the area of the region $$\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$$. Then $${1 \over 2}\left( {\Delta  - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)$$ is equal to

Accepted Answer

Required area $=2 \int\limits_{1}^{3} 2 \sqrt{x} d x+\int_{3}^{\sqrt{21}} \sqrt{\left(21-x^{2}ight)} d x$

$=2\left(\left[\left.2\left(\frac{x^{3 / 2}}{3 / 2}ight)ight|_{1} ^{3}ight]+\left[\frac{x}{2} \sqrt{21-x^{2}}+\frac{21}{2} \sin ^{-1}\left(\frac{x}{\sqrt{21}}ight)ight]_{3}^{\sqrt{21}}ight)$

$=2 \sqrt{3}+\frac{21 \pi}{2}-\frac{8}{3}-21 \sin ^{-1} \sqrt{\frac{3}{7}}=\Delta$

$	herefore \frac{1}{2}\left(\Delta-21 \sin ^{-1}\left(\frac{2}{\sqrt{7}}ight)ight)=\sqrt{3}-\frac{4}{3}$

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