9. Average Value of a Function by Integration

by M. Bourne

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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?

Continues below

Example

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

T = 0.001t4 − 0.280t2 + 25

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

What was the average temperature during the day?

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What have the following got in common?

  • The Average Value of a Function
  • The Area under a Curve
  • A bag of air
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