The area of the region R=(x, y): |x| ≤ y and x^2 + y^2 ≤1, in square…

Let A be the required area. Then,

A = 2 (Area of the shaded region in first quadrant)

$⇒A=2\int\limits_{0}^{1/\sqrt{2}}(y_1-y_2)dx$

$⇒A=2\int\limits_{0}^{1/\sqrt{2}}(\sqrt{1-x^2}-x)dx$

$⇒A=2\left[\frac{1}{2}x\sqrt{1-x^2}+\frac{1}{2}\sin^{-1}x-\frac{x^2}{2}\right]_{0}^{1/\sqrt{2}}$

$⇒A=2\left[\frac{1}{2\sqrt{2}}×\frac{1}{\sqrt{2}}+\frac{1}{2}×\frac{п}{4}-\frac{1}{4}\right]=\frac{п}{4}$ sq. units.