Area Question 18
Question 18
- The area (in sq units) 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
(2014 Main) (a) $\frac{\pi}{2}+\frac{4}{3}$ (b) $\frac{\pi}{2}-\frac{4}{3}$ (c) $\frac{\pi}{2}-\frac{2}{3}$ (d) $\frac{\pi}{2}+\frac{2}{3}$
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Solution:
- Given, $A=\left{(x, y): x^{2}+y^{2} \leq 1\right.$ and $\left.y^{2} \leq 1-x\right}$
Required area $=\frac{1}{2} \pi r^{2}+2 \int_{0}^{1}\left(1-y^{2}\right) d y$
$$ \begin{aligned} & =\frac{1}{2} \pi(1)^{2}+2 \quad y-{\frac{y^{3}}{3}}_{0}^{1} \ & =\frac{\pi}{2}+\frac{4}{3} \text { sq units } \end{aligned} $$
