Introduction to :

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].

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Exercise 1

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

Exercise 2

Let f (x) = x. Find a constant m such that
Answer
Here's the graph of f (x) = x on [-2,4]:
From this graph we can see that
We want
m(4-(-2)) = 6.
Since (4-(-2)) = 6, We must have m = 1. The weighted area between f (x) = x and the x-axis on [-2,4] is the same as the weighted area between m = 1 and the x-axis on [-2,4].

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]
Answer
The average value of f (x) = x on [-10,0] is
Since
the average value of f (x) = x on [-10,0] is
Since f (x) is below the x-axis on all of [-2,2], it's reassuring that we got a negative number for the average value.

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]
Answer
The function on [-2,2] describes half a circle.
The area of this half-circle is
so
The average of f on [-2,2] is
Since f (x) is below the x-axis on all of [-2,2], it's reassuring that we got a negative number for the average value.

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]
Answer
The average value of f ( x ) = sin x on [-π,π] is
Since sin is an odd function, its integral on [-π,π] is 0. So the average value of f (x) = sin x on [-π, π] is

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]
Answer
The graph of f (x) = 3x + 2 on [1,4] describes a trapezoid with width 3 and heights 5 and 14.
The area of this trapezoid is
so
The average of f on [1,4] is
.

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]
Answer
The line -2x – 1 hits the x-axis when
so the triangle above the x-axis has base 3.5 and the triangle below the x-axis has base 2.5:
Then
We conclude that the average of f on [-4,2] is

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