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Introduction to The Fundamental Theorem Of Calculus - At A Glance:

The average value of the function f on the interval [a,b] is the integral of the function on that interval divided by the length of the interval. Since we know how to find the exact values of a lot of definite integrals now, we can also find a lot of exact average values. What's the average value of an "A" in Calculus class? You tell us.

Sample Problem

Find the average value of f(x) = sin x on the interval .

Answer.

The average value of f(x) = sin x on this interval is

Since we know how to evaluate the integral, we know how to find the average value. First let's simplify that stuff out in front of the integral:

Now we can rewrite the average value to be a little more tidy.

It's tempting to go off and compute the integral in a corner of your paper, then come back and multiply by  at the end. Unfortunately, that's dangerous. After working out a long integral, it's very easy to forget to come back and do that last step. Don't do it.

Example 1

Find the average value of f(x) = x2 on the interval [0,2].


Exercise 1

Find the average value of the function on the indicated interval.

g(t) = cos t on 

Exercise 2

Find the average value of the function on the indicated interval.

f(x) = 2x + 5 on [1,4]

Exercise 3

Find the average value of the function on the indicated interval.

f(x) = ex on [0,1]

Exercise 4

Find the average value of the function on the indicated interval.

s(t) = 7t ln 7 on [1,3]

Exercise 5

Find the average value of the function on the indicated interval.

 on [1,3]

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