Common Mistakes
Calculus AB · AP · Applications of Integration
Focus
Using Accumulation Functions and Definite Integrals in Applied Contexts
This topic is assessed on the AP Exam.
Frequent mistakes
### 1. Integral setup mismatch
What it looks/sounds like: Integrand or bounds do not match the story.
Why it happens: Formula is memorized without labeling quantities.
How to fix: Write what ∫ represents (net change, area, volume) before evaluating.
### 2. Volume: missing π or wrong slice
What it looks/sounds like: Disk/washer integral omits π or uses wrong radius.
Why it happens: Slice thickness variable does not match bounds.
How to fix: Draw the slice; label R, r, and dx or dy.
### 3. Distance vs displacement
What it looks/sounds like: Total distance is computed with a single signed integral.
Why it happens: Velocity sign is ignored.
How to fix: Split intervals or use |v(t)| for total distance.
### 4. Correct work, wrong question
What it looks/sounds like: The algebra is fine but answers a different quantity than the stem asks for.
Why it happens: Attention locks onto the first operation instead of the final ask.
How to fix: Underline the final question; after solving, verify the boxed answer matches it.
### 5. Notation and justification gaps
What it looks/sounds like: Missing units, limit language, or ‘because’ statements AP graders expect.
Why it happens: Notation is treated as decoration rather than part of the score.
How to fix: Add one justification phrase per major step (definition, theorem, or rule).
How to avoid them
1. Sketch the region before writing the integral. 2. State what the integral represents. 3. Keep dx/dy consistent with bounds.
Quick self-check
- [ ] Can set up a volume integral
- [ ] Can distinguish displacement and distance
- [ ] Can label a diagram for area/volume