AP Calculus AB / Integration and Accumulation of Change
Lesson
Adding slices: Riemann sums to definite integrals
The integral is the limit of sums of (height)(width).
Learning goals
- Build a left or right Riemann sum from a table or graph.
- Interpret a definite integral as net change of a quantity.
Explanation
Cut [a, b] into n pieces. On each piece, pick a sample height f(x_i*) and multiply by the width Δx. Add. That is a Riemann sum. As the widths shrink, the sum approaches the definite integral when f is reasonably tame. Left sums use the left edge of each slice; right sums use the right.
If v(t) is velocity in meters per second, ∫ from 0 to 5 of v(t) dt is net displacement in meters, not necessarily distance traveled. Distance needs the speed |v|. Signed area below the axis subtracts. Units are the clue that you added a rate across time.
Key terms
- Riemann sum. A sum of sampled heights times widths that approximates a definite integral.
- Definite integral. The signed accumulation of f from a to b, written ∫_a^b f(x) dx.
Common mistakes
- Using function values as if they were already areas.
- Calling net displacement “distance” when the velocity changed sign.
Practice
Original Marlow Works items. Check the answer explanation after you try.
Water enters a tank at a rate r(t) = 8 − 0.5t liters per minute for 0 ≤ t ≤ 10. At t = 0 the tank holds 20 liters. (a) Write an expression for the volume V(t). (b) Find V(6). (c) Interpret V(6) − V(0) in context.
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
f is positive and decreasing. Is a left Riemann sum an over- or underestimate of ∫_a^b f?
Answer. An overestimate, because each left height is taller than the function on that slice.
The rectangles stick up above a falling graph.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference 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
Open source