Net change and the area between two graphs

Subtract the lower function. Split the integral if they swap places.

Learning goals

Compute the average value of a function on [a, b].
Set up an area between two curves with the correct order.

Explanation

If a quantity’s rate is r(t), the net change from a to b is the integral of r. The average value of f on [a, b] is that integral divided by the length b − a. It is a mean height, the constant function that would accumulate the same area.

Area between y = f(x) and y = g(x) is ∫ (top − bottom) dx on the interval where those labels hold. If the curves cross, split the integral. Sometimes x = x(y) is kinder, and you integrate (right − left) dy. The sketch decides the algebra.

Key terms

Average value. The number (1/(b − a)) ∫_a^b f(x) dx, the mean height of f on the interval.
Net change. The definite integral of a rate, equal to F(b) − F(a) when F′ is that rate.

Common mistakes

Forgetting to split an area integral where the upper curve changes.
Calling a signed integral “area” when the integrand is negative.

Practice

Original Marlow Works items. Check the answer explanation after you try.

1 of 2Limits and Continuity · easy · multiple choice

What is lim (x→3) of (x² − 9) / (x − 3)?

Original Marlow Works item — not a College Board question.

Take your time—this is practice, not a test.

Answer explanation

f has average value 6 on [2, 5]. What is ∫_2^5 f(x) dx?

Answer. 6 × (5 − 2) = 18.

Average value times interval length recovers the integral.

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Related resources

External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.

Reference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
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