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Evaluate $$\iint_{D}(x+2 y) d A,$$ where $$D$$ is the region bounded by the
parabolas $$y=2 x^{2}$$ and $$y=1+x^{2}$$

$$ {\int_{x_{1}=-1}^{x_{2}=1}}_{R}  \left[\int_{y_{1}=2x^{2}}^{y_{2}=x^{2}+1}(x+2 y)d y\right] d x $$

$$=\int_{x_{1}=-1}^{x_{2}=1} \int_{2 x^{2}}^{x^{2}+1}\left[{xy+y^{2}}\right] d x=\int_{-1}^{1} \left[x\left(x+1-2 x^{2}\right)+\left(x^{2}+1\right)^{2}-\left(2 x^{2}\right)^{2}\right]$$

=$$\int_{-1}^{1} \left(x-x^{3}\right)+x^{4}+2 x^{2}+1-4 x^{4}$$

$$\int_{-1}^{1} -3 x^{4}-x^{3}+2 x^{2}+x+1$$

$$=\frac{32}{15} $$

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