8.1 Average Value of a Function

The formula

The average value of f on [a, b] is (1/(b − a))·∫[a to b] f(x) dx — the definite integral divided by the interval length.

Meaning

It's the height of a rectangle over [a, b] with the same area as the region under f. The average value of a constant c is just c.

Key terms

Average value

(1/(b − a))·∫[a to b] f dx.

Divide by

The interval length (b − a).

Meaning

Height of an equal-area rectangle.

Constant f

Its average value is itself.

Uses

A definite integral over the interval width.

Try an AP-style question

What is the average value of f(x) = x² on [0, 3]?

1. 3
2. 9/2
3. 4.5
4. 9