The area of the region described by A $= \{(x, y): x^{2} + y^{2} \le 1$ and $y^{2} \le 1 - x\}$ is$: (1) \pi/2 - \frac{2}{3} (2) \pi/2 + \frac{2}{3} (3) \pi/2 + \frac{4}{3} (4) \pi/2 - \frac{4}{3}$

The area of the region described by A = {(x, y): x² + y² ≤ 1 and y² ≤ 1 - x} is: $x^2 + y^2 \le 1 \quad (\text{inside the unit circle})$ $y^2 \le 1 - x$ $\text{Find the area of } A$ $C: x^2 + y^2 = 1, \quad P: y^2 = 1 - x \ (\text{vertex } (1,0), \text{ opens left})$ $x^2 + (1-x) = 1 \Rightarrow x(x-1) = 0 \Rightarrow x = 0, 1$ $\text{Points: } (1,0), \ (0,1), \ (0,-1)$ $y^2 \le 1 - x \iff x \le 1 - y^2 \ (\text{left of the parabola})$ $A = \text{unit disk minus the two caps (circle} \leftrightarrow \text{parabola)}$ $\text{Upper cap: } \int_0^1\left[\sqrt{1-x^2} - \sqrt{1-x}\right]dx = \frac{\pi}{4} - \frac{2}{3}$ $\text{Caps (both)} = 2\left(\frac{\pi}{4} - \frac{2}{3}\right) = \frac{\pi}{2} - \frac{4}{3}$ $\text{Area}(A) = \pi - \left(\frac{\pi}{2} - \frac{4}{3}\right) = \frac{\pi}{2} + \frac{4}{3}$ $\text{Area}(A) = \frac{\pi}{2} + \frac{4}{3} \Rightarrow \text{option } (3)$