Average Value of a Function
Mean value of a continuous function over an interval.
Formula
Variables
Example
Area divided by width
How it works
This formula gives the average output of a continuous function across an interval, not the output at a single point. It comes from the idea of replacing the graph by a rectangle with the same area: the integral measures total accumulated value, and dividing by the interval length converts that total into a per-unit average. In effect, it is the constant height that would produce the same signed area over the interval.
When to use it
Use it when you need the long-run average of a quantity that changes continuously, such as temperature, speed, pressure, or concentration over time or distance. It is also useful in physics and engineering for averaging signals or rates over an interval.
Worked example, step by step
Find the average value of f(x) = x^2 on the interval [1, 4]. Step 1: Write the formula: \bar{f} = \frac{1}{4-1}\int_1^4 x^2\,dx Step 2: Compute the integral: \int x^2\,dx = \frac{x^3}{3} Step 3: Evaluate from 1 to 4: \int_1^4 x^2\,dx = \frac{4^3}{3} - \frac{1^3}{3} = \frac{64-1}{3} = \frac{63}{3} = 21 Step 4: Divide by the interval length: \bar{f} = \frac{1}{3}(21) = 7 Final answer: The average value of f(x) on [1, 4] is 7.
Common mistakes to avoid
- Forgetting to divide by the interval length b-a after integrating
- Using the antiderivative value without evaluating it between a and b
- Confusing average value with the value of the function at the midpoint or endpoints
Did You Know?
The average value of a function is the height of the rectangle with the same area — the mean of infinitely many points.
Share this formula
More in Calculus
View allPower Rule (Derivative)
IntermediateDerivative of a power function.
Chain Rule
IntermediateDerivative of a composite function.
Product Rule
IntermediateDerivative of a product of two functions.
Quotient Rule
IntermediateDerivative of a quotient of two functions.
Derivative of Sine
IntermediateDerivative of the sine function.
Derivative of Cosine
IntermediateDerivative of the cosine function.