Calculus I · Unit 3A · lesson
Average Value of a Function
Learn Average Value of a Function through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Riemann sums and the definite integralWhat this section is building
Learn Average Value of a Function through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.
Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.
Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.
Learning objectives
Calculate and interpret the average value of a continuous function over an interval.
Average Value of a Function
A continuous average is total contribution divided by width
For finitely many numbers, an average is their sum divided by how many values were added. For a continuous function, the integral replaces the sum and the interval length replaces the count. Thus the average value is the constant height that would produce the same total accumulation over the interval.
This interpretation is often more useful than the formula alone. Average temperature over a day, average power over a time interval, and average density along a rod all use the same idea. For a continuous function, the average-value theorem guarantees that the function actually attains this average somewhere in the interval, although it may do so at more than one point and the theorem does not tell us where without further work.
The average of finitely many numbers is their sum divided by the count. For continuously distributed values, the integral plays the role of the sum and interval length plays the role of the count:
Average electrical power
If instantaneous power is watts on , then
The oscillating part contributes zero net average over a full period.
u3a-average-value-01Find the average value of on .
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Integrate, then divide by interval length 2.
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Read this graph as text
Equal-area average-height rectangle. A curve and a rectangle over the same interval have equal area; rectangle height is the average value. Compare the area under a curve with a rectangle of equal area and height f avg. Do not confuse average function value with average rate of change.
Every relationship in equal-area average-height rectangle uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Compare the area under a curve with a rectangle of equal area and height f avg.
Equal-area average-height rectangle. Compare the area under a curve with a rectangle of equal area and height f avg.
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