Question

The area of the region 
\(\begin{array}{l} \left\{\left(x,y\right);\left|x-1\right|\leq y \leq \sqrt{5-x^2} \right\}\end{array}\)
is equal to

• A. \(\frac{5}{2}sin^{−1}⁡(\frac{3}{5})−\frac{1}{2}\)
• B. \(\frac{5π}{4}−\frac{3}{2}\)
• C. \(\frac{3π}{4}+\frac{3}{2}\)
• D. \(\frac{5π}{4}−\frac{1}{2}\)

Answer 1

The Correct Option is D

\(\begin{array}{l} A=\displaystyle\int\limits_{-1}^1\left(\sqrt{5-x^2}-\left(1-x\right)\right)dx+\displaystyle\int\limits_{1}^2\left(\sqrt{5-x^2}-\left(x-1\right)\right)dx\end{array}\)
\(\begin{array}{l} \left.A=2\left(\frac{x}{2}\sqrt{5-x^2}+\frac{5}{2}\sin^{-1}\frac{x}{\sqrt{5}}\right)-2x\right|_0^1 \left.+\frac{x}{2}\sqrt{5-x^2}+\frac{5}{2}\sin^{-1}\frac{x}{\sqrt{5}}-\frac{x^2}{2}+x\right|_1^2\end{array}\)
\(\begin{array}{l} =\left(\frac{5\pi}{4}-\frac{1}{2}\right)\text{ sq. units} \end{array}\)