Question

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

• A. $\sqrt{3}-\frac{2}{3}$
• B. $2 \sqrt{3}-\frac{2}{3}$
• C. $2 \sqrt{3}-\frac{1}{3}$
• D. $\sqrt{3}-\frac{4}{3}$

Answer 1

Answer: (D)

Area $2 \int\limits_1^3 2 \sqrt{x} d x+2 \int\limits_3^{\sqrt{21}} \sqrt{21-x^2 d x}$
$ \Delta=\frac{8}{3}(3 \sqrt{3}-1)+21 \sin ^{-1}\left(\frac{2}{\sqrt{7}}\right)-6 \sqrt{3} $
$ \frac{1}{2}\left(\Delta-21 \sin ^{-1}\left(\frac{2}{\sqrt{7}}\right)\right)=\frac{2 \sqrt{3}-\frac{8}{3}}{2} $
$ =\sqrt{3}-\frac{4}{3}$