# 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?

### 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.

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