The area bounded by $y = |x| - 1$ and $y = 1 - |x|$ is ______
Step-by-Step Solution
Key Concept: The area between two curves is found by integrating the difference of the upper and lower curves over the interval of intersection.
The two curves are $y = |x| - 1$ and $y = 1 - |x|$. These intersect where $|x| - 1 = 1 - |x|$, giving $2|x| = 2$, so $|x| = 1$ at points $(-1, 0)$ and $(1, 0)$. The curve $y = 1 - |x|$ is above $y = |x| - 1$ in the interval $[-1, 1]$. By symmetry, the area is $2\int_0^1 [(1 - x) - (x - 1)] dx = 2\int_0^1 (2 - 2x) dx = 2[2x - x^2]_0^1 = 2(2 - 1) = 2 \times 2 = 4$ sq. units.
Correct Answer: 4