What is the area of the region (in the first quadrant) bounded by y=\sqrt{1-x^{2}}, y=x and y=0 ?
- A. \frac{\pi}{4}
- B. \frac{\pi}{6}
- C. \frac{\pi}{8} ✓
- D. \frac{\pi}{12}
Correct Answer: C. \frac{\pi}{8}
Explanation
The equation y = \sqrt{1-x^2} represents the upper half of the unit circle x^2 + y^2 = 1 with radius 1. The region is in the first quadrant, bounded above by the circle, on the left by the line y=x (which makes an angle of \pi/4 with the x-axis), and on the bottom by the x-axis (y=0). This geometric shape is a circular sector with radius r=1 and central angle \theta = \pi/4. Its area is \frac{1}{2}r^2\theta = \frac{1}{2}(1)^2(\frac{\pi}{4}) = \frac{\pi}{8}.
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