Determine the average value of $f(x)=x^2$ between $0$ and $2$.
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Lesson Parent:
Problem:
Determine the average value of $f(x)=x^2$ between $0$ and $2$.
Answer:
$4/3$
Solution:
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 $2$:
