Question

The area of the region $\left\{(x,y):|x-1|\le y\le \sqrt{5-x^2}\right\}$ is equal to :

• A. $\frac{5}{2}{\sin }^{-1}\left(\frac{3}{5}\right)-\frac{1}{2}$
• B. $\frac{5\pi }{4}-\frac{3}{2}$
• C. $\frac{3\pi }{4}+\frac{3}{2}$
• D. $\frac{5\pi }{4}-\frac{1}{2}$

Answer 1

Answer: (D)

$|x-1|<y<\sqrt{5-x^2}$
When $|x-1|=\sqrt{5-x^2}$
$\Rightarrow (x-1{)}^2=5-x^2$
$\Rightarrow x^2-x-2=0$
$\Rightarrow x=2,-1$
Required Area $=$ Area of $△ABC+$ Area of region $BCD$
$=\frac{1}{2}\left|\begin{array}{ccc}1& 0& 1\\ 2& 1& 1\\ -1& 2& 1\end{array}\right|+\frac{\pi }{4}(\sqrt{5}{)}^2-\frac{1}{2}(\sqrt{5}{)}^2$
$=\frac{5\pi }{4}-\frac{1}{2}$