Answer
$2$
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:
$A=\int_a^b [f(x)-g(x)] \ dx\\ = \int_{0}^{1} [1-y^2-(2y^2-2)] \ dy \\=\int_0^1 (3-3y^2) \ dy \\=[3y-y^3]_0^1\\=(3)(1)-1-0 \\=2$