Answer
The area of the shaded region is $22/15$.
Work Step by Step
The shaded region is bounded above by the curve $y=x^2$ and below by the curve $y=-2x^4$ and runs from $x=-1$ to $x=1$.
Therefore, the area of the shaded region is $$A=\int^{1}_{-1}(x^2+2x^4)dx$$ $$A=\Big(\frac{x^3}{3}+\frac{2x^5}{5}\Big)\Big]^{1}_{-1}$$ $$A=\Big(\frac{1}{3}+\frac{2}{5}\Big)-\Big(-\frac{1}{3}-\frac{2}{5}\Big)$$ $$A=2\Big(\frac{1}{3}+\frac{2}{5}\Big)$$ $$A=2\Big(\frac{11}{15}\Big)=\frac{22}{15}$$
