Average Function Value

The first application of integrals which we'll see is the average value of a function. The given fact tells us how to calculate this.

Average Function Value

The average value of function f ( x ) over the interval [a,b] is specified by,

                                               f_avg = 1/(b-a)     ∫^b_af ( x ) dx

Let's work on some quick examples.

Example Find out the average value of following functions on the specified interval.

f (t ) = t ² - 5t + 6 cos (∏, t ) on  [-1, 5/2 ]

Solution

There's actually not a lot to do in this problem other than just utilizes the formula.

f _avg      =  1/   ((5/2)-(-1)∫^(5/2)_(-1)  t² - 5t + 6cos( ∏ t) dt|_(-1)^(5/2)

           = (2/7)((1/3)t3-(5/2)t2+(6/ ∏)sin(∏ t) |_(-1)^(5/2)

            = 12/7 ∏ - 13/6

            = -1.620993

You caught the substitution required for the third term right?

Therefore, the average value of this function of the given interval is -1.620993.