Step 1: Understanding the Concept:
The region is the intersection of the interior of a unit circle \( x^2 + y^2 = 1 \) and the region to the left of the parabola \( x = 1 - y^2 \).
Step 2: Key Formula or Approach:
The total area can be divided into a semi-circular part (for \( x<0 \)) and a parabolic segment part (for \( x \geq 0 \)).
Step 3: Detailed Explanation:
For \( x<0 \), the region is the left half of the circle \( x^2 + y^2 = 1 \).
\[ \text{Area}_1 = \frac{1}{2} \pi(1)^2 = \frac{\pi}{2} \]
For \( x \geq 0 \), we look at the area bounded by \( y^2 \leq 1 - x \), which means \( x \leq 1 - y^2 \). This is a parabola opening to the left with vertex at \( (1, 0) \).
The intersections of the circle and parabola at \( x \geq 0 \) are at \( x=0 \).
Total area for \( x \geq 0 \) bounded by \( x = 1 - y^2 \) and the y-axis:
\[ \text{Area}_2 = \int_{-1}^{1} (1 - y^2) dy = [y - \frac{y^3}{3}]_{-1}^1 \]
\[ \text{Area}_2 = (1 - \frac{1}{3}) - (-1 + \frac{1}{3}) = \frac{2}{3} + \frac{2}{3} = \frac{4}{3} \]
Total Area \( = \text{Area}_1 + \text{Area}_2 = \frac{\pi}{2} + \frac{4}{3} \).
Step 4: Final Answer:
The total area is \( \pi/2 + 4/3 \).