1 Average value of a function
Suppose a time-varying function [maths rendering] is defined on the interval [maths rendering] . The area, [maths rendering] , under the graph of [maths rendering] is given by the integral [maths rendering] . This is illustrated in Figure 5.
Figure 5
On Figure 3 we have also drawn a rectangle with base spanning the interval [maths rendering] and which has the same area as that under the curve. Suppose the height of the rectangle is [maths rendering] . Then
area of rectangle = area under curve [maths rendering]
The value of [maths rendering] is the mean value of the function across the interval [maths rendering] .
Key Point 2
The mean value of a function [maths rendering] in the interval [maths rendering] is [maths rendering]
The mean value depends upon the interval chosen. If the values of [maths rendering] or [maths rendering] are changed, then the mean value of the function across the interval from [maths rendering] to [maths rendering] will in general change as well.
Example 2
Find the mean value of [maths rendering] over the interval [maths rendering] .
Solution
Using Key Point 2 with [maths rendering] and [maths rendering] and [maths rendering]
[maths rendering]
Task!
Find the mean value of [maths rendering] over the interval [maths rendering] .
Use Key Point 2 with [maths rendering] and [maths rendering] to write down the required integral:
[maths rendering]
Now evaluate the integral:
[maths rendering]