AP Calculus AB and BC glossary

Average value of a function

Also called: Mean value of a function

The average value of a function on an interval is the definite integral over that interval divided by the length of the interval. It is the constant height a rectangle would need to have the same area as the region under the curve.

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx

The rectangle picture is the useful one: you are flattening the region under the curve into a rectangle of the same width and equal area, and the average value is its height.

The mistake

Averaging the endpoint values, or forgetting to divide by b−ab - a. Dividing by the interval length is what makes it an average rather than a total.

This is the continuous form of expected value. Where a discrete random variable weights each outcome by its probability and sums, a continuous one integrates against a density, and StatsLearn works the discrete version in expected value.

Appears in: Unit 8: Applications of Integration