C34 · Unit C8 Applications of integration
Displacement and total distance
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 8.2
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
Integrating velocity gives displacement; integrating speed |v| gives total distance. Locate every velocity sign change and split the interval before removing the absolute value. Position at the final time adds the displacement to initial position. An acceleration integral gives velocity change and likewise requires an initial velocity for an absolute value of velocity.
Worked example
For v(t) equals t minus 2 on [0,5], displacement is [t squared /2 minus 2t] sub 0 to the power 5 equals 2.5. Distance is minus ∫ sub 0 squared (t minus 2)dt+∫ sub 2 to the power 5 (t minus 2)dt equals 2+4.5 equals 6.5. The distance exceeds the magnitude of displacement because the direction changes.
Explore the mathematics
Displacement and total distance
Separate signed displacement from total distance.
Scroll sideways to see the full diagram.
Predict before continuing
Predict whether displacement and total distance agree.
Read the complete explanation
- Predict whether displacement and total distance agree.
- Velocity is negative until t=1 and positive afterward.
- The signed integral is 0, but distance integrates |v|.
- Distance is 1/2+1/2=1; displacement is 0. Split at each sign change.
The velocity changes sign at t=2. The negative triangle has area magnitude 2 and the positive one 4.5. Displacement is −2+4.5=2.5; distance is 2+4.5=6.5.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
Total distance cannot be negative. Taking the absolute value after integrating may hide backtracking.
Investigate and explain
Draw a signed area diagram under velocity and physically trace the motion along a number line.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.