What is the area of the region between the graphs of $y=x^2$ and $y=-2x$ from $x=1$ to $x=3$ ?

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The graph of $y=x^2$ has a positive value from $x=1$ to $x=3$ whereas the graph of $y=-2x$ is always negative on the same interval.

The region bounded by the curves can be represented by:

$$\int\limits_1^3 \colorbox{aqua}{$x^2$}-\colorbox{yellow}{$(-2x)$}   dx$$ $$= \int\limits_1^3 x^2+2x   dx$$ $$= \Big[\frac{1}{3}x^3+x^2+C \Big]_1^3$$ $$= \frac{1}{3}(3)^3+(3)^2-\left(\frac{1}{3}(1)^3+(1)^2\right)$$ $$= 18-\frac{4}{3}$$ $$= \boxed{\frac{50}{3}}$$