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Calculus I · Unit 3A · lesson

Average Value of a Function

Concept

Learning objectives

Calculate and interpret the average value of a continuous function over an interval.

Average Value of a Function

Explanation

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 1b−a∫abf(x) dx\frac{1}{b-a}\int_a^b f(x)\,dx 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:

favg=1b−a∫abf(x) dx.f_{\mathrm{avg}}=\frac1{b-a}\int_a^bf(x)\,dx.
Application

Average electrical power

If instantaneous power is P(t)=100+20sin⁡tP(t)=100+20\sin t watts on [0,2π][0,2\pi], then

Pavg=12π∫02π(100+20sin⁡t) dt=100 W.P_{\mathrm{avg}}=\frac1{2\pi}\int_0^{2\pi}(100+20\sin t)\,dt=100\text{ W}.

The oscillating part contributes zero net average over a full period.

Interactive checku3a-average-value-01

Find the average value of f(x)=2xf(x)=2x on [1,3][1,3].

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Integrate, then divide by interval length 2.

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Equal-area average-height rectangle. Compare the area under a curve with a rectangle of equal area and height f avg.
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.

Visual study

Equal-area average-height rectangle. Compare the area under a curve with a rectangle of equal area and height f avg.

After the explanation

Use the section idea

Reading lens

Treat a definite integral as the limit of structured approximations, with the sample rule and sign visible in every rectangle.

Mental model

Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.

Decision

Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.

Common trap

Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.

Check yourself

Can you construct the sum from a table or formula and predict whether it is high or low before calculating?

Continue

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