# 9. Average Value of a Function by Integration

by M. Bourne

The average value of the function *y* = *f*(*x*) from *x = a* to *x* = *b* is given by:

`y_"ave"=(int_a^bf(x)dx)/(b-a`

**Why?** When you see a formula like this for the first time, think about where it comes from and why it should work.

Hint: How do we find the average of a set of numbers? What are we really doing each time we find an integral? What does the integral symbol stand for?

### Example

### Need Graph Paper?

The temperature *T* (in °C) recorded during a day followed the curve

T =0.001t^{4}− 0.280t^{2}+ 25

where *t* is the number of hours from noon (-12 ≤ *t* ≤ 12)

What was the average temperature during the day?

Answer

First, we consider the graph of the situation and estimate that the average should be around 14 to 16 degrees.

The graph of temperature `T` at time `t`.

`T_["ave"]=(int_a^bf(x) text[d]x)/(b-a)`

`=(int_-12^12(0.001t^4-0.28t^2+25)\ dt)/(12-(-12))`

`=1/24[(0.001t^5)/(5)-(0.28t^3)/(3)+25t]_-12^12`

`=2/24[(0.001t^5)/(5)-(0.28t^3)/(3)+25t]_0^12`

`=1/12[49.7664-161.28+300]`

`=15.7 ^@ text[C]`

Our earlier estimate was quite reasonable.

### Question

What have the following got in common?

- The Average Value of a Function
- The Area under a Curve
- A bag of air
- The most famous car crash in history
- Boxers

All is revealed in Head Injury Criterion.