Integrating Vector-Valued Functions
Integrate components with separate constants and the appropriate initial velocity and position.
College Board Topic 9.5 · CED p. 170 (PDF p. 175) · CED effective Fall 2020; project course outline 2026–27.
Recovering position from velocity is the same accumulation idea as one-dimensional motion, once for each component. If and , then . Integrate the x- and y-components separately. Each indefinite component integral has its own constant; one shared scalar constant would force an unjustified relationship between the coordinates.
If acceleration is given, integrate once to obtain velocity and use the initial velocity. Integrate again to obtain position and use the initial position. Keep these two initial conditions separate. A rate vector without an initial position describes many parallel translations of the same path.
A definite vector integral is displacement, not a final position until you add the starting position. Numerical component integration is allowed when formulas are unwieldy, but state the supplied initial time and position. If a finite table is used, identify the chosen rule and call the result an estimate.
Let's work through it
- Given and , integrate to . Why: Each component is an independent antiderivative.
- Use t=1: , . Why: The initial position fixes both integration constants.
- Thus , and . Why: Evaluate the position formula, not merely the displacement.
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
See every recall question with its answer
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Retrieval 1
Worked answer and reason
Integrate separately and apply both initial coordinates.
Retrieval 2
Worked answer and reason
Two integrations introduce constants at both stages.
Catch the tempting mistake
- The integral of velocity is automatically position.
- A definite velocity integral is displacement. Add the given initial position; indefinite integration needs component constants.
Calculator workspace and limits
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Put it to work
- Unit 9 problem 13 · No calculator · analytic
- Unit 9 problem 14 · No calculator · analytic
- Unit 9 problem 15 · No calculator · numerical
For the teacher
- Require students to substitute the initial condition back into their answer. That catches most constant errors immediately.
Glossary and deeper reading · 6 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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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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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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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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Initial condition
An initial condition specifies a solution’s value at a particular input. It can determine the constant in a family of solutions to a differential equation. For example, y′=2x with y(0)=3 selects y=x²+3. The solution must satisfy both the equation and the stated condition on a suitable interval.
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Vector in calculus
A vector records a magnitude and direction and can be represented by ordered components. In planar motion, the components describe changes or rates along the horizontal and vertical axes. For example, velocity ⟨3,4⟩ has speed 5 because its magnitude is √(3²+4²). A vector-valued function lets those components vary with a parameter.
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Net change
Net change is the final amount minus the initial amount. Integrating a rate gives this signed change when the model and integral are valid. For example, moving 5 meters right and then 2 meters left gives net displacement 3 meters. Opposite contributions cancel, so net change can differ from the total amount of movement or activity.