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Average Value of a Function

Objectives

  • Describe an average value of a dataset
  • Demonstrate the relation with integration and average value
  • Describe an average value of a function
  • Show a visual representation of the Average Value Theorem

Summary

One of the most valuable applications of the definite integral is that it provides a way to discuss the average value of a function, even for a function that takes on infinitely many values. Recall that if we wish to take the average of n numbers y 1 , y 2 , … , y n , we compute

A V G = y 1 + y 2 + ⋯ + y n n .

Since integrals are essentially a sum of all of the possible f ( x ) values, it should not be surprising that we can use the integral to compute the average value over some interval ( b − a ) .

A V G = ∫ a b f ( x ) d x b − a .

Alternatively

Another way to interpret the definite integral: the definite integral of a function f from a to b is the length of the interval ( b − a ) times the average value of the function on the interval. In addition, when the function f is nonnegative on [ a , b ] , the average value has a natural visual representation.

average value theorem
F i g . 1

Consider F i g . 1 , where we see at left the shaded region whose area is ∫ a b f ( x ) d x , at center the shaded rectangle whose dimensions are ( b − a ) ⋅ AVG ⁡ [ a , b ] . The Average Value Theorem tells us that the area of the blue region in the left figure is the same as the area of the green rectangle in the center figure. Thus, knowing the average value of a function enables us to construct a rectangle whose area is the same as the value of the definite integral of the function on the interval.

See the Desmos demonstration on Average Value of a Function. 

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