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Question

Mathematics Question on applications of integrals

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:

A

π2+43\frac{\pi}{2} + \frac{4}{3}

B

π2−43\frac{\pi}{2} - \frac{4}{3}

C

π2−23\frac{\pi}{2} - \frac{2}{3}

D

π2+23\frac{\pi}{2} + \frac{2}{3}

Answer

π2+43\frac{\pi}{2} + \frac{4}{3}

Explanation

Solution

Shaded area
=π(1)22+2∫01(1−x)dx=\frac{\pi(1)^{2}}{2}+2 \int\limits_{0}^{1} \sqrt{(1-x)} d x
=π2+2(1−x)3/23/2(−1)∣01=\frac{\pi}{2}+\left.\frac{2(1-x)^{3 / 2}}{3 / 2}(-1)\right|_{0} ^{1}
=π2+43(0−(−1))=\frac{\pi}{2}+\frac{4}{3}(0-(-1))
=π2+43=\frac{\pi}{2}+\frac{4}{3}