Connecting Position, Velocity, and Acceleration of Functions Using Integrals
Later connection: recover motion from rates and distinguish displacement from distance.
Before this connection: Accumulation and definite integrals; Antiderivatives and integration tools.
College Board Topic 8.2 · CED p. 148 (PDF p. 153) · CED effective Fall 2020; project course outline 2026–27.
For motion along a line, s′=v and v′=a. Thus s(b)=s(a)+∫ₐᵇ v(t)dt and v(b)=v(a)+∫ₐᵇ a(t)dt. The integral of velocity is displacement; the integral of speed |v| is total distance. Include the initial position or velocity when the question asks for a final value.
A particle changes direction where velocity changes sign, not just where acceleration is zero. Find velocity zeros, check signs, and split the distance integral there. If only a table is given, use an approximation and state the sampling assumptions. Units also distinguish the quantities: velocity integral gives length, acceleration integral gives velocity, and the total distance cannot be negative.
Let's work through it
- Let v(t)=2t−2 on [0,3]. It changes sign at t=1. Reason: Direction is determined by velocity sign.
- Displacement is [t²−2t]₀³=3. Reason: Integrating velocity gives signed position change.
- Distance is −∫₀¹ v+∫₁³ v=1+4=5. Reason: Speed takes the absolute value on each sign interval.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Does ∫v=0 force total distance to be zero?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Does ∫v=0 force total distance to be zero?
Worked answer and reason
No.
Forward and backward displacement can cancel.
Retrieval 2
If acceleration is negative and velocity is negative, is the particle speeding up?
Worked answer and reason
Yes.
Velocity and acceleration have the same sign, so speed increases.
Catch the tempting mistake
- Use the sign of acceleration to decide forward or backward motion.
- Direction comes from velocity. Acceleration describes how velocity changes.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 8 problem 3 · No calculator · verbal
- Unit 8 problem 4 · No calculator · analytic
- Unit 8 problem 5 · No calculator · graphical
For the teacher
- Require a sign chart before distance calculations. Have students annotate the same motion in position, velocity, and verbal representations.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
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Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
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Acceleration in calculus
Acceleration is the derivative of velocity with respect to time. In one-dimensional motion, it is also the second derivative of position. For example, s(t)=t² meters gives constant acceleration 2 meters per second squared. A negative acceleration can increase speed when the velocity is also negative.
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Total distance traveled
Total distance traveled adds the lengths of every part of a motion path without canceling opposite directions. In one dimension it is the integral of speed, |v(t)|, over the time interval. For example, moving 5 meters right and then 2 meters left gives total distance 7 meters. When velocity changes sign, split the integral at those sign changes or integrate its absolute value correctly.