Determine the average value of $f(x)=x^2$ between $0$ and $2$.

To determine the average value we need to solve
$$\frac{1}{2-0}\int_0^2x^2\,dx.$$

Now the antiderivative of $f(x)=x^2$ is $F(x)=x^3/3$ so we have
$$\begin{eqnarray} \frac{1}{2-0}\int_0^2x^2\,dx&=&\frac{1}{2}\left(\frac{2^3}{3}-\frac{0}{3}\right)\\ &=&\frac{1}{2}\cdot \frac{2^3}{3}\\ &=&\frac{2^2}{3}\\ &=&\frac{4}{3}=1.333. \end{eqnarray}$$

Here is a picture of the average of $x^2$ between $0$ and $2$: