Answer
$\dfrac{10}{3}$
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:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{1} (4x-x^2) \ dx+\int_{1}^{2} (4x-x^2(-4x-4)] \ dx \\=[2x^2-\dfrac{x^3}{3}]_0^{1}+[4x-\dfrac{x^3}{3}]_1^2 \\=[2(1)^2-\dfrac{1^3}{3}-0]+[4(2)-\dfrac{(2)^3}{3}-[4(1)-\dfrac{(1)^3}{3}]\\=\dfrac{10}{3}$
