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Definite Integrals

Definite Integrals

At a Glance - Averages with Functions

Sample Problem

Let f(x) be the function graphed below.

We can see that 

Find a constant m such that

In other words, find a constant m so that we have this:

Answer. . In order to have this equal 16, m must be 2.

We call this constant m the average value of f on [a, b]. When we take the integral of f on [a, b], we get some number. This number is like the sum of all the test scores: it's the accumulation of all the stuff.

To average that accumulation we give every x the same function value as every other x. Therefore we end up with a constant function whose integral on [a, b] is the same as the integral of f on [a, b].

The average value of f on [a, b] is a y-value. It's the particular y-value for which the weighted area between that y-value and the x-axis is equal to the integral of f on [a, b]. The average value of f on [a, b] is the (weighted) height of the rectangle whose (weighted) area is equal to the integral of f on [a, b].

Let f be non-negative for the sake of the pictures and let m be the average value of f on [a, b]. The area under m is a rectangle. Whatever area is in that rectangle but not under f must make up for the area that is under f but not part of the rectangle.

Example 1

Let f (x) = 4 – x2. Is the average value of f on [-2, 2]

  • strictly between 0 and 2?
  • equal to 2?
  • strictly between 2 and 4?
  • equal to 4?

Example 2

Calculate the average value of  on [0, 3].


Exercise 1

Let f (x) = 3x. Find a constant m such that


Exercise 2

Let f (x) = x. Find a constant m such that


Exercise 3

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  f (x) = x on [-10, 0]

Exercise 4

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  on [-2, 2]

Exercise 5

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  • f (x) = sin x on [-π, π]

Exercise 6

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  • f (x) = 3x + 2 on [1, 4]

Exercise 7

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  f (x) = -2x – 1 on [-4, 2]

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