To solve the given problem, we need to find the area of the region bounded by the functions $f(x) = |x^2 - 1|$ and $g(x) = -x^2$. Let’s break it into steps:

Step 1: Understand the Functions

1. $f(x) = |x^2 - 1|$:

   • The absolute value splits the function into two cases:
     • When $x^2 - 1 \geq 0$ (i.e., $x \leq -1$ or $x \geq 1$): $f(x) = x^2 - 1$,
     • When $x^2 - 1 < 0$ (i.e., $-1 < x < 1$): $f(x) = -(x^2 - 1) = 1 - x^2$.

2. $g(x) = -x^2$:

   • This is a simple downward-opening parabola.

Step 2: Find Intersection Points

To determine the bounds of the region, find where $f(x)$ and $g(x)$ intersect.

1. Case 1: $f(x) = x^2 - 1$ intersects $g(x) = -x^2$ for $x \leq -1$ or $x \geq 1$: $x^2 - 1 = -x^2$ Simplify: $2x^2 = 1 \implies x^2 = \frac{1}{2} \implies x = \pm\frac{\sqrt{2}}{2}.$

2. Case 2: $f(x) = 1 - x^2$ intersects $g(x) = -x^2$ for $-1 < x < 1$: $1 - x^2 = -x^2$