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