Answer
The area of the shaded region is $128/15$.
Work Step by Step
The shaded region is bounded above by the curve $y=2x^2$ and below by the curve $y=x^4-2x^2$ and runs from $x=-2$ to $x=2$.
Therefore, according to the definition, the area of the shaded region is $$A=\int^{2}_{-2}\Big(2x^2-(x^4-2x^2)\Big)dx$$ $$A=\int^{2}_{-2}(4x^2-x^4)dx$$ $$A=\frac{4x^3}{3}\Big]^{2}_{-2}-\frac{x^5}{5}\Big]^{2}_{-2}$$ $$A=\frac{4}{3}\Big(8-(-8)\Big)-\frac{1}{5}\Big(32-(-32)\Big)$$ $$A=\frac{64}{3}-\frac{64}{5}=\frac{128}{15}$$
