Calculus AB · Applications of Integration · 14 min read · Updated 2026-05-10
Position, velocity, acceleration via integration — AP Calculus AB
AP Calculus AB · Applications of Integration · 14 min read
1. Recovering Velocity and Position with Initial Conditions★★☆☆☆⏱ 4 min
We start from the fundamental derivative relationships you already know: velocity is the rate of change of position, and acceleration is the rate of change of velocity. Integration reverses differentiation, so we can integrate acceleration to get velocity, then integrate velocity to get position, using initial conditions to solve for unknown constants of integration.
v(t) = \frac{ds}{dt}, \quad a(t) = \frac{dv}{dt}
v(t) = \int a(t) dt + C \\ s(t) = \int v(t) dt + D
Exam tip: Always label your constants $C$ and $D$ separately when integrating twice, and solve for each constant as soon as you have the required initial condition. Don't wait to solve for both at the end, which often leads to substitution errors.
2. Net Displacement vs. Total Distance Traveled★★★☆☆⏱ 5 min
Once you have the velocity function $v(t)$, you can calculate two distinct quantities over an interval $[a,b]$ that are frequently confused on the AP exam. Net displacement is the net change in position from start to end of the interval, accounting for direction, while total distance is the total length of the path traveled, regardless of direction.
To calculate total distance, all motion counts as positive, so we integrate the absolute value of velocity. First find all points in $[a,b]$ where $v(t)=0$ (direction changes), split the integral at these points, adjust the sign for intervals where $v(t)$ is negative, then add the results.
Exam tip: Always check that the total distance is greater than or equal to the absolute value of net displacement. If your calculation gives the opposite, you made a sign error when splitting the integral.
3. Finding Values at a Point with the Fundamental Theorem★★☆☆☆⏱ 3 min
A common AP exam problem asks for velocity or position at a specific time, given an initial value, but does not require the full function. In these cases, you can use the Fundamental Theorem of Calculus directly to avoid solving for constants of integration, saving time and reducing errors.
Exam tip: If the question only asks for a value of velocity/position at a specific time, skip finding the full function and use the Fundamental Theorem form directly to save 1–2 minutes on exam day.
4. Concept Check★★★☆☆⏱ 2 min
Common Pitfalls
Why: Students remember to split the integral at zeros of $v(t)$ but just drop the absolute value without checking the sign, treating $|v(t)|=v(t)$ everywhere.
Why: Students confuse the constants, using the same $C$ for both integration steps and solving once instead of solving for a new constant after the second integration.
Why: Students confuse net displacement with total distance, especially when the question says 'how far did the particle travel' which can seem ambiguous to new learners.
Why: Students solve for all roots automatically and split at all roots, even ones outside the interval, leading to extra unnecessary subintervals and calculation errors.
Why: Students mix up which initial condition goes with which integration step.