question_answer

Area of the region enclosed between the curves \[x={{y}^{2}}-1\] and \[x=\,\,|y|\sqrt{1-{{y}^{2}}}\] is

A) 1

B) 4/3

C) 2/3

D) 2

Correct Answer: D

Solution :

\[\because \]       \[x={{y}^{2}}-1,\]       \[x=\,\,|y|\sqrt{1-{{y}^{2}}}\]

\[\therefore \] \[=2\,\,\left[ \int\limits_{0}^{1}{y\sqrt{1-{{y}^{2}}}}dy+\left| \int\limits_{0}^{1}{({{y}^{2}}-1})\,dy \right| \right]\]\[=2\,\,\left[ \frac{1}{3}+\left| -\frac{2}{3} \right| \right]=2\,\,\left[ \frac{1}{3}+\frac{2}{3} \right]=2\]