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

(A) $\frac{\pi}{2}+\frac{4}{3}$

(B) $\frac{\pi}{2}-\frac{4}{3}$

(C) $\frac{\pi}{2}-\frac{2}{3}$

(D) $\frac{\pi}{2}+\frac{2}{3}$

JEE Main · 2014 · 06 Apr 2014 · Offline · Area Under Curves

The area of the region described by

A = {(x, y) : x^{2} + y^{2} \leq 1

and

y^{2} \leq 1 - x}

Answer: (A)

The region is hemmed in by the unit disc

x^{2} + y^{2} \leq 1

and the parabola

y^{2} \leq 1 - x

, that is

x \leq 1 - y^{2}

. To the left of the

y

-axis the disc lies entirely inside the parabola, contributing the left half-disc of area

\frac{π}{2}

. To the right, between

x = 0

and

x = 1

, the parabola forms the boundary, adding

2 \int_{0}^{1} 1 - x d x = 2 \cdot \frac{2}{3} = \frac{4}{3}

. The total area is

\frac{π}{2} + \frac{4}{3}

JEE Main · 2014 · 06 Apr 2014 · Offline · Single correct option