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}_{a}f ( 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) } t^{2 }- 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.