To find the area of the region bounded by the equations \(1 - y^2 = |x|\) and \(|x| + |y| = 1\), we need to analyze these curves in detail and determine the points of intersection and the shape of the region.
Step 1: Understanding the curves
- The equation \(1 - y^2 = |x|\) represents a pair of parabolas. The regions are symmetric about the x-axis, with the parabolas opening to the left and right from \((0, 1)\) and \((0, -1)\) as vertices.
- The equation \(|x| + |y| = 1\) describes a square in the coordinate plane centered at the origin with vertices at \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\).
Step 2: Points of intersection
- Substitute \(|x| = 1 - y^2\) into \(|x| + |y| = 1\) to find points of intersection.
- So, \((1 - y^2) + |y| = 1\) simplifies to \(|y| = y^2\).
- This implies \(y = 0\) or \(y = 1\) and \(y = -1\), respectively.
- Thus, the points of intersection are \((0, 1)\) and \((0, -1)\) along the y-axis.
Step 3: Calculating the area
- The region of interest is inside the square and between the parabolas. The given symmetry and positions imply symmetry about the x-axis and y-axis.
- To find the area of one quadrant and then multiply by 4: Integrate within the region using only one portion, such as from \(y = 0\) to \(y = 1\).
- For \(y \in [0, 1]\), find x-limits using the parabola: \(x = 1 - y^2\).
The area covered in one section from \(y = 0\) to \(y = 1\):
- Integrate \(x = 1 - y^2\) for \(y \in [0, 1]\):
- \(\int_{0}^{1} (1 - y^2) \, dy = \left[y - \frac{y^3}{3}\right]_{0}^{1} = 1 - \frac{1}{3} = \frac{2}{3}\)
Since the region is symmetric and occurs in four quadrants, the total area is \(\frac{2}{3} \times 1 = \frac{2}{3}\) sq units, seeing the entire covered region of the bounded space.
Conclusion: The total area of the bounded region is \(\frac{2}{3}\) square units, which matches the correct answer.