The Fundamental Theorem of Calculus
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.
Find the average value of f(x) = sin x on the interval .
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.