- 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
Introduction to :
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.
Practice:
Find the average value of f(x) = x2 on the interval [0,2]. |
Find the average value of the function on the indicated interval.
g(t) = cos t on 
along the way.
Find the average value of the function on the indicated interval.
f(x) = 2x + 5 on [1,4]
Find the average value of the function on the indicated interval.
f(x) = ex on [0,1]
Find the average value of the function on the indicated interval.
s(t) = 7t ln 7 on [1,3]
Find the average value of the function on the indicated interval.
on [1,3]
.
