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

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Aicharya · Accepted Answer

Region: $\left\{(x, y) \mid x^2+y^2 \leq 1\right.$ and $\left.y^2 \leq 1-x\right\}$

Required area $\begin{aligned} & A=2\left[\int_{-1}^0 \sqrt{1-x} d x+\int_0^1 \sqrt{1-x} d x\right] \\ & =2\left[\frac{x}{2} \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1} x\right]_{-1}^0+2\left[\frac{-2}{3}(1-x)^{\frac{3}{2}}\right]_0^1 \\ & =\frac{\pi}{2}+\frac{4}{3} \end{aligned}$

The area of the region described by $\left\{(x, y) / x^2+y^2 \leq 1\right.$ and $\left.y^2 \leq 1-x\right\}$…

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