What It Means to Integrate a Vector-Valued Function
A vector-valued function describes more than one quantity at a time. In two-dimensional motion, a position function is commonly written as
where and give the horizontal and vertical coordinates of the particle.
The key idea is simple: integrate each component separately. If you know the velocity vector, you can integrate its horizontal component to obtain the horizontal position function and integrate its vertical component to obtain the vertical position function.
The constant of integration is itself a vector. An initial position, such as , determines the constants in the two components.
The Formula
If the vector-valued function is
then its indefinite integral is
In other words, integrate the components one at a time. There is no special "vector integration rule" beyond applying ordinary integration to each component.
Why This Matters for Motion Problems
For a moving particle, the relationships among position, velocity, and acceleration are exactly the same as in one-dimensional calculus, except that each quantity has multiple components.
So if an AP problem gives you acceleration and an initial velocity, you can integrate acceleration to find velocity. If it then gives you an initial position, you can integrate velocity to find position.
Think of the process as moving backward through the derivative chain:
A Step-by-Step Method
- Write the vector in component form. Separate the and components.
- Integrate each component independently. Use ordinary antiderivative rules.
- Add a constant to each component. The constants are determined by initial conditions.
- Use the initial vector condition. Substitute the given time and position or velocity values.
- Check by differentiating. Your answer should return the original vector function.
Worked Examples
Find an antiderivative of .
A particle moves in the plane with velocity . At time , its position is . Find the position function .
A particle has acceleration and initial velocity . Find its velocity function.
A particle has velocity . Find its displacement from to .
A particle has acceleration , initial velocity , and initial position . Find its position function.
Definite Integrals and Displacement
One of the most useful applications is finding displacement directly from velocity. If a particle has velocity vector , then its displacement from time to time is
This works component by component:
Notice the distinction between displacement and distance traveled. The vector integral gives displacement. It does not automatically give the total distance traveled along the path.
Position from an Initial Position and Velocity
Sometimes an AP problem gives velocity and an initial position rather than asking you to find an indefinite integral. In that case, a definite integral can be especially efficient.
This formula says: new position = starting position + accumulated velocity.
For example, if and , then
This approach automatically incorporates the initial condition, so you do not have to introduce arbitrary constants.
Common Mistakes
A vector has multiple components. If , you must integrate both and . Forgetting one component gives an incomplete answer.
Each component gets its own constant. If , the general antiderivative is , not an expression with only one constant shared automatically between the components.
If you are given velocity and integrate it, you obtain position. If you are given acceleration and integrate it once, you obtain velocity—not position. You need to integrate acceleration twice to obtain position.
The integral of the velocity vector gives displacement. Total distance requires integrating the speed, which is the magnitude of the velocity vector: .
If a problem gives an initial position or initial velocity, use it. The initial condition is what determines the constants of integration and turns a family of functions into one specific motion.
Practice Problems
Find an antiderivative of .
Show answer
A particle has velocity and position at . Find .
Show answer
A particle has acceleration and initial velocity . Find .
Show answer
A particle has velocity . Find its displacement from to .
Show answer
A particle has acceleration , initial velocity , and initial position . Find its position function.
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon