The average value of a function tells you the constant height that would give the same total accumulation over an interval. In calculus, this connects a graph, an area, and a real-world average in one idea. It matters because many quantities change continuously, such as temperature, velocity, population rate, and power use.
Instead of averaging a few data points, an integral averages all values across the interval.
Key Facts
- Average value formula: f_avg = (1 / (b - a)) ∫_a^b f(x) dx
- Total accumulated quantity: ∫_a^b f(x) dx = f_avg(b - a)
- For velocity v(t), average velocity over [a, b] is v_avg = (1 / (b - a)) ∫_a^b v(t) dt
- If f(x) is continuous on [a, b], then its average value exists.
- Mean Value Theorem for Integrals: if f is continuous on [a, b], then there is some c in [a, b] such that f(c) = f_avg.
- Average value has the same units as f(x), while ∫_a^b f(x) dx has units of f times x.
Vocabulary
- Average value
- The constant value of a function that produces the same total area or accumulation over an interval.
- Definite integral
- A calculation that gives the net accumulated value of a function between two input values.
- Interval length
- The distance between the endpoints of an interval, equal to b - a for [a, b].
- Accumulation
- The total amount built up by a changing rate or quantity over an interval.
- Average height rectangle
- A rectangle with width b - a and height f_avg whose area equals the area under the function.
Common Mistakes to Avoid
- Forgetting to divide by b - a. The integral gives total accumulation, not the average value.
- Using (a + b) / 2 as the average value. That only finds the midpoint of the interval, not the average output of the function.
- Averaging endpoint values only. The expression (f(a) + f(b)) / 2 is not generally equal to the calculus average value.
- Ignoring units. If f(t) is speed in meters per second and t is in seconds, the integral is meters, but the average value is meters per second.
Practice Questions
- 1 Find the average value of f(x) = x^2 on the interval [0, 3].
- 2 A car has velocity v(t) = 4t + 10 meters per second for 0 ≤ t ≤ 5. Find its average velocity over the interval.
- 3 A temperature function rises quickly in the morning and slowly later in the day. Explain why averaging only the starting and ending temperatures may give a different result from the integral average value.