Common Mistakes
Calculus BC · AP · Integration and Accumulation of Change
Focus
The Fundamental Theorem of Calculus and Definite Integrals
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. Missing +C
What it looks/sounds like: +C is omitted on an indefinite integral.
Why it happens: Antiderivative is treated as unique.
How to fix: Add +C unless evaluating a definite integral with limits.
### 3. FTC limit reversal
What it looks/sounds like: Reversed limits do not change the sign of the integral.
Why it happens: Bounds are swapped without negating.
How to fix: Track sign when swapping ∫ₐᵇ to ∫ᵦₐ.
### 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. State what the integral accumulates (with units). 2. Show FTC evaluation steps clearly. 3. Distinguish net change from total area when needed.
Quick self-check
- [ ] Can write a definite integral for net change
- [ ] Can apply FTC correctly
- [ ] Can include +C when required