Area between a cubic and a parabola

Text

Compute the area of the region between the curves $y=x^{3}+4$ and $y=x^{2}$ for $1\le x\le 4$.

Solution

On $[1,4]$ we have $x^{3}+4>x^{2}$, so $A=\displaystyle\int_1^4\big[(x^{3}+4)-x^{2}\big]dx=\Big[\tfrac{x^{4}}{4}+4x-\tfrac{x^{3}}{3}\Big]_1^4=\dfrac{176}{3}-\dfrac{47}{12}=\dfrac{657}{12}=54.75$. $\boxed{A=\tfrac{657}{12}=54.75}$