Question:
The area of the region
\[
R = \{(x, y) \mid |x| \le |y| \text{ and } x^2 + y^2 \le 1\}
\]
is:
The area of the region
\[ R = \{(x, y) \mid |x| \le |y| \text{ and } x^2 + y^2 \le 1\} \]
is:
\[ R = \{(x, y) \mid |x| \le |y| \text{ and } x^2 + y^2 \le 1\} \]
is:
Show Hint
Convert inequalities to angular regions in polar coordinates.
Updated On: Mar 23, 2026
- (3π)/(8) sq units
- (5π)/(8) sq units
- (π)/(2) sq units
- (π)/(8) sq unit
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The Correct Option is A
Solution and Explanation
The condition \(|x| \le |y|\) corresponds to angles between
\[ \theta = \frac{\pi}{4} \text{ to } \frac{3\pi}{4} \] and similarly in all four quadrants.
Total angular region:
\[ 4 \times \frac{\pi}{4} = \pi \]
Area inside the unit circle:
\[ \frac{\pi}{2\pi} \times \pi = \frac{3\pi}{8} \]
\[ \theta = \frac{\pi}{4} \text{ to } \frac{3\pi}{4} \] and similarly in all four quadrants.
Total angular region:
\[ 4 \times \frac{\pi}{4} = \pi \]
Area inside the unit circle:
\[ \frac{\pi}{2\pi} \times \pi = \frac{3\pi}{8} \]
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