Area Between Curves Question 1

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

$\dfrac{2}{3}$

$12$

$\dfrac{50}{3}$

$\dfrac{58}{3}$

Approach

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:

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