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Introduction to Definite Integrals - At A Glance:

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], you 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.

  1.  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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