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The area of the region described by $A= \left\{(X,Y):x^{2}+y^{2}\leq 1 \right\} and \left\{y^{2}\leq 1-x\right\}$ is

$\frac{\pi}{2}+\frac{4}{3}$

$\frac{\pi}{2}-\frac{4}{3}$

$\frac{\pi}{2}-\frac{2}{3}$

$\frac{\pi}{2}+\frac{2}{3}$

Option A

Given, $A= \left\{(X,Y):x^{2}+y^{2}\leq 1 \right\} and \left\{y^{2}\leq 1-x\right\}$ is

Required area = $\frac{1}{2}\pi r^{2}+2\int_{0}^{1} (1-y^{2})dy$

= $\frac{1}{2}\pi (1)^{2}+2\left(y-\frac{y^{3}}{3}\right)_0^1$

= $\frac{\pi}{2}+\frac{4}{3}$

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