Find the area bounded by the graphs of two functions 

First find the intersection points of the two curves: 
\[y=x^2,    y=x^3   \Rightarrow x^2=x^3   \]
\[x^3-x^2=0    \Rightarrow   x^2(x-1)=0    \Rightarrow x=0, 1.\]
Note that for $x$ between $0$ and $1$, we have $x^2 >x^3$. Thus the area bounded between the two graphs is 
\[A=\int_0^1 x^2-x^3 dx=(\frac{x^3}{3}-\frac{x^4}{4})|_0^1=(\frac{1}{3}-\frac{1}{4})=\frac{1}{12}.\]