question_answer

The      area      of     the      region\[\{(x,y):{{x}^{2}}+{{y}^{2}}\le 1\le x+y\}\]is:

A)  \[\frac{{{\pi }^{2}}}{5}sq\,unit\]                              

B)         \[\frac{{{\pi }^{2}}}{2}sq\,unit\]

C)         \[\frac{{{\pi }^{2}}}{3}sq\,unit\]                              

D)         \[\frac{\pi }{4}sq\,unit\]

E)         \[\left( \frac{\pi }{4}-\frac{1}{2} \right)sq\,unit\]

Correct Answer: E

Solution :

Required area \[=\frac{1}{4}(Area\text{ }of\text{ }circle)-(Area\text{ }of\,\Delta AOB)\] \[=\frac{\pi }{4}-\frac{1}{2}\times OA\times OB\] \[=\left( \frac{\pi }{4}-\frac{1}{2} \right)sq\,unit\]