The average value of a function tells you the typical height of a curve over an interval. Instead of averaging a list of numbers, calculus averages all the function values from x = a to x = b. This matters because many real quantities, such as velocity, temperature, and power, change continuously over time.
The average value gives one constant number that represents the same total accumulation over the interval.
The key idea is to compare the signed area under the curve y = f(x) with the area of a rectangle. The rectangle has width b - a and height f_avg, so its area is (b - a)f_avg. Setting this equal to the definite integral from a to b gives f_avg = (1/(b - a))∫_a^b f(x) dx.
If f is continuous, the Mean Value Theorem for Integrals guarantees that at least one point c in [a,b] has f(c) = f_avg.
Key Facts
- Average value formula: f_avg = (1/(b - a))∫_a^b f(x) dx.
- The interval length is b - a, and it must be positive, so b > a.
- The definite integral ∫_a^b f(x) dx represents signed area, with area below the x-axis counted as negative.
- The equal-area rectangle has area (b - a)f_avg.
- For a constant function f(x) = k, the average value is f_avg = k.
- Mean Value Theorem for Integrals: if f is continuous on [a,b], then there is at least one c in [a,b] such that f(c) = f_avg.
Vocabulary
- Average value
- The single constant value that gives the same signed area over an interval as the original function.
- Definite integral
- A quantity that measures the net signed area between a function and the x-axis over a specified interval.
- Interval
- The set of x-values from a starting point a to an ending point b, written [a,b].
- Signed area
- Area above the x-axis counted as positive and area below the x-axis counted as negative.
- Mean Value Theorem for Integrals
- A theorem stating that a continuous function reaches its average value at least once on a closed interval.
Common Mistakes to Avoid
- Forgetting to divide by b - a. The integral gives total signed accumulation, not the average height.
- Using b + a instead of b - a for the interval length. The width of the interval is the distance from a to b, so it is b - a.
- Treating all area as positive when the function goes below the x-axis. The average value formula uses signed area, so negative portions reduce the result.
- Assuming f_avg must occur at the midpoint of the interval. The Mean Value Theorem for Integrals guarantees at least one point c, but it does not have to be the midpoint.
Practice Questions
- 1 Find the average value of f(x) = 2x + 1 on the interval [0,4].
- 2 Find the average value of f(x) = x^2 on the interval [1,3].
- 3 A continuous function has average value 5 on [2,8]. Explain what this tells you about the area under the curve and what the Mean Value Theorem for Integrals guarantees.