Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) f(x) dx means the definite integral of f(x) from x = a to x = b.
Recap
- The average value of a continuous function f on [a, b] is the integral divided by the length of the interval.
- It comes from averaging many equally spaced samples of f: the sample average is a Riemann sum divided by b − a.
- Picture: the average value is the height of the rectangle on [a, b] with the same signed area as the region under f.
- If f is continuous, f reaches its average value at least once in [a, b] (Intermediate Value Theorem). That c need not be the midpoint, and need not be unique.
- The average value can be negative. It is usually not the average of the end values, (f(a) + f(b))/2. That shortcut is guaranteed only when f is linear.
Key relationships
| Quantity | Formula | Units |
|---|---|---|
| Average value of f | (1/(b − a)) ∫ (a to b) f(x) dx | units of f |
| Total (net) accumulation | ∫ (a to b) f(x) dx = f_avg × (b − a) | units of f × units of x |
| Average rate of change of f | (f(b) − f(a))/(b − a) | units of f per unit of x |
| Average value of f′ | equals the average rate of change of f | units of f per unit of x |
From a table: estimate the integral with a Riemann or trapezoidal sum (weight each value by its width), then divide by b − a. From a graph: add signed areas, then divide by b − a.
Assumptions behind the method
- f is continuous on [a, b] (or at least integrable, for the formula; continuity is needed for “f reaches its average”).
- The interval is the one in the question, not the whole table or graph.
- A table estimate is only as good as the sum you use; say which one.
Mistakes to avoid
- Forgetting 1/(b − a). The integral alone is a total.
- Confusing average value with average rate of change.
- Averaging table entries with unequal gaps.
- Wrong units, such as °C·hours for an average temperature.
- Ignoring signs: area below the axis counts as negative.
- Assuming c is the midpoint of the interval.
Quick self-check
- Find the average value of 6x² on [0, 2]. (8: the integral is 16, divided by 2)
- The average value of f on [3, 8] is 7. Find ∫ (3 to 8) f(x) dx. (35 = 7 × 5)
- Find the average value of 2x + 1 on [0, 4], and say why it equals (f(0) + f(4))/2 here. (5. The function is linear, so the region under it is a trapezoid, whose area is width × average of the two end heights.)
Next: practice questions.