Answer
$\pi-2$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{-1}^{1} [\dfrac{2}{1+x^2}-1] \ dx\\=[2 \arctan x-x]_{-1}^1 \\=2 \arctan (1)-1- [-2 \arctan (-1) -(-1)] \\=2\times \dfrac{\pi}{4}-1-[2(\dfrac{-\pi}{4})+1] \\=\pi-2$