Question

The area of the region \(R=\{(x,y):|x|\le |y| \text{ and x^2+y^2\le1\}\) is

• A. \(\dfrac{3\pi}{8}\) sq units
• B. \(\dfrac{5\pi}{8}\) sq units
• C. \(\dfrac{\pi}{2}\) sq units
• D. \(\dfrac{\pi}{8}\) sq unit

Hint:

Convert inequalities into angular limits.

Answer 1

The Correct Option is A

Step 1: Condition \(|x|\le |y|\) implies region between lines \(y=\pm x\).
Step 2: In polar coordinates, this corresponds to angles: \[ \theta\in\left[\frac{\pi}{4},\frac{3\pi}{4}\right]\cup \left[\frac{5\pi}{4},\frac{7\pi}{4}\right] \]
Step 3: Area inside unit circle: \[ A=2\times\frac12\left(\frac{3\pi}{4}-\frac{\pi}{4}\right)=\frac{3\pi}{8} \]