- Topics At a Glance
- Integrating the Velocity Function
- Negative Velocity
- Change in Position
- Total Distance Travelled vs. Change in Position
- About the Fundamental Theorem of Calculus (FTC)
**Using the FTC to Evaluate Integrals**- Integrating with Letters
- Order of Limits of Integration
- Why the Choice of Antiderivative Doesn't Matter
**Average Values**- Units
- Word Problems
- The Second Fundamental Theorem of Calculus
- Letters
- Antiderivatives
- Finding Derivatives
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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 .

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 |

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

Exercise 3

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

*f*(*x*) = *e ^{x}* on [0,1]

Exercise 4

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

*s*(*t*) = 7* ^{t}* ln 7 on [1,3]

Exercise 5

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

on [1,3]