Chapter 6.3, Problem 14E

To determine

To calculate the integral of the given function using average mean value theorem.

Answer to Problem 14E

  $x=0,x=2$

Explanation of Solution

Given:

The given function is $y={(x-1)}^2$ and $\left[0,3\right]$

Calculation:

If f is integrated on $\left[a,b\right]$ , its average (mean) value on $\left[a,b\right]$ is

  $av(f)=\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$

So,

  $\begin{array}{l}av({(}^x)=\frac{1}{3-0}{\displaystyle \underset{0}{\overset{3}{\int }}{(x-1)}^{2}dx}\\ =\frac{1}{3}{\displaystyle \underset{0}{\overset{3}{\int }}{(x-1)}^{2}dx}\end{array}$

Now, calculate the integral by using NINT.

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{3}{\int }}{(x-1)}^{2}dx=\text{NINT}\left({(x-1)}^2,x,0,3\right)}\\ \approx 3\end{array}$

So,

  $\begin{array}{l}av\left({(x-1)}^2\right)\approx \frac{1}{3}\times 3\\ \approx 1\end{array}$

Substitute the average value of ${(x-1)}^2$ in equation of y.

  $\begin{array}{l}$ 1={(x-1)}^2$x-1=\pm 1\\ x=0\\ \text{and}\\ x=2\end{array}$