Average value
The average value of f on [a, b] is the integral of f over [a, b], divided by the length b - a.
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Analogy
Levelling out a bumpy water surface until it is flat: the flat height is the average.
Example
Integral from 0 to 3 is 9.Length of interval is 3.Average = 9 / 3 = 3.It lies between 0 and 9, as it should.
In practice
Gives mean temperature over a day, or mean speed over a trip, from a continuous formula.
Practice
Flashcards
- What is the average value of f on [a, b]?The integral of f from a to b, divided by b - a.
Questions people ask
Why divide by b minus a?
The integral is a total, like total distance. Dividing a total by the length of the interval gives the average, in the same way that dividing total distance by total time gives average speed. Without dividing, you would get a sum, not a typical value.
Does the average value always occur at some point?
Yes, if f is continuous. The mean value theorem for integrals says that f reaches its average at some point c in the interval. For x squared on [0, 3], the average is 3, reached at x equal to the square root of 3.