AP® Calculus AB · Unit 8 · Topic 8.1
The Mean Value Theorem for Integrals
The Mean Value Theorem for Integrals says any continuous function on [a,b] actually equals its own average value at some point c inside the interval.
A continuous function can't skip over values. If it dips below its average somewhere and rises above it somewhere else, it has to cross its own average value in between, since it never jumps. That crossing point is the c the theorem promises: f(c) equals the average value of f on [a,b], where average value is defined as (1/(b-a)) times the integral of f from a to b.
The equal-area picture makes this concrete. The integral of f from a to b gives the exact area under the curve. The average value, multiplied by (b-a), gives the area of a rectangle with that same width and height equal to the average. Both areas match by definition. Since f is continuous, somewhere it must actually reach the height of that rectangle, otherwise the curve's area couldn't equal the rectangle's area. That point is c, and f(c) is the rectangle's height.
Keep this separate from the Mean Value Theorem for derivatives, which finds a point where the instantaneous slope matches the average slope over an interval. Here you're matching a function value to an average value, not a slope to an average slope. They share a name and a continuity requirement, but one deals in areas and one deals in tangent lines. This topic also sets up the rest of the unit: once you can find where a velocity function equals its average, you're one step from computing total displacement, since displacement is just that average velocity times elapsed time, which is exactly the integral itself.
Key terms
- Average value of a function
- The quantity (1/(b-a)) times the integral of f from a to b, representing the constant height with the same total area.
- Mean Value Theorem for Integrals
- For f continuous on [a,b], there exists c in [a,b] where f(c) equals the average value of f on that interval.
- Equal-area rectangle
- A rectangle with width (b-a) and height equal to the average value, whose area matches the area under the curve.
- Continuity
- The property of having no breaks or jumps, which guarantees the function must pass through its average value.
Watch it explained
- How it worksHow does the equal-area rectangle picture prove the Mean Value Theorem for integrals?
- Worked exampleFind the c where f(x) = √x hits its average value on the interval 0 to 4.
- Common mistakeWhy isn't this theorem the same as the Mean Value Theorem for derivatives?
- Ties togetherHow does the average value of a velocity function relate to total displacement?
Check yourself
End of topic.