First we need to know what is. When we look at the graph, we see a trapezoid with heights 6 and 12 and width 2.
The area of this trapezoid is
In order for to be 18 also, m must be 9.
Example 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].
Example 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.
Example 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.
Example 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 sine is an odd function, its integral on [-π, π] is 0. So the average value of f (x) = sin x on [-π, π] is
.
Example 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
.
Example 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 can conclude that the average of f on [-4, 2] is
is. When we look at the graph, we see a trapezoid with heights 6 and 12 and width 2.
to be 18 also, m must be 9.
on [-2, 2]
on [-2, 2] describes half a circle.
.
