Question

The area of the region $\{(x,y): x^{2} + y^{2} \le 1 \le x + y\}$ is

• A. $\dfrac{\pi^{2}}{5}$ sq unit
• B. $\dfrac{\pi^{2}}{2}$ sq unit
• C. $\left(\dfrac{\pi}{4} - \dfrac{1}{2}\right)$ sq unit
• D. $\dfrac{\pi}{4}$ sq unit

Hint:

Sketch the region first. Use geometric formulas (sector, triangle) rather than integration when the shapes are standard.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The region is the part of the unit disk lying above (or on) the line $x+y=1$.
Step 2: Detailed Explanation:
The unit circle $x^2+y^2=1$ and line $x+y=1$ intersect at $(1,0)$ and $(0,1)$.
Area $=$ (quarter circle area) $-$ (area of triangle with vertices $(0,0),(1,0),(0,1)$)
$= \dfrac{\pi(1)^2}{4} - \dfrac{1}{2}(1)(1) = \dfrac{\pi}{4} - \dfrac{1}{2}$.
Step 3: Final Answer:
Area $= \dfrac{\pi}{4} - \dfrac{1}{2}$ sq units.