Area between a parabola and a line

Text

Compute the area of the region between the parabola $y=x^{2}$ and the line $y=3x$.

Solution

Intersections: $x^{2}=3x\Rightarrow x=0,\ x=3$. The line is above on this interval. $A=\displaystyle\int_0^3 (3x-x^{2})dx=\Big[\tfrac{3x^{2}}{2}-\tfrac{x^{3}}{3}\Big]_0^3=\dfrac92=4.5$. $\boxed{A=\tfrac92=4.5}$