Area of the region bounded by the given curves.

Find the area of the bounded region by the curves y=x^3 and y=x^2.

 

first solve simultaneously to find the end points

 

x^3=x^2

x=0 or x=1

I can also see by simple inspection that x^2 and x^3 will be greater or equal to zero in this domain

and x^2 will be above x^3

 

so

 

 

\(\displaystyle\int_{0}^{1}\;\;x^2-x^3\;\;dx \\ =\left[\frac{x^3}{3}-\frac{x^4}{4}\right]_0^1\\ =\frac{1}{3}-\frac{1}{4}\\ =\frac{1}{12} \;\;units ^2\)