Practice a College Board-style free response question on Riemann sums, accumulation, and the Fundamental Theorem of Calculus. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 6 · Integration & Accumulation
Water flows into a tank at a rate of R(t) = 6t − t² gallons per hour, for 0 ≤ t ≤ 6 hours. The tank contains 4 gallons at time t = 0.
A
Using a right Riemann sum with the three subintervals [0, 2], [2, 4], [4, 6], approximate the total amount of water that flows in during the 6 hours.
✓ Model answer
Δt = 2, with right endpoints t = 2, 4, 6. R(2) = 12 − 4 = 8; R(4) = 24 − 16 = 8; R(6) = 36 − 36 = 0. Right sum = 2(8) + 2(8) + 2(0) = 16 + 16 + 0 = 32 gallons.
Why it scores: Uses Δt = 2 and the correct right-endpoint values of R, then sums. An arithmetic setup with the wrong endpoints or width loses points.
B
Find the exact total amount of water that flows into the tank during the 6 hours by evaluating a definite integral.
✓ Model answer
Total inflow = ∫ from 0 to 6 of (6t − t²) dt = [3t² − t³/3] from 0 to 6 = (3·36 − 216/3) − 0 = (108 − 72) = 36 gallons.
Why it scores: Correct antiderivative 3t² − t³/3, correctly evaluated via the FTC to 36. A missing term or evaluation slip loses the point.
C
Let W(t) be the amount of water in the tank at time t. Write an expression for W(t) using an integral, and find the amount of water in the tank at t = 6.
✓ Model answer
Since W accumulates from the initial 4 gallons: W(t) = 4 + ∫ from 0 to t of R(x) dx. At t = 6, W(6) = 4 + 36 = 40 gallons (using the total inflow of 36 from part B).
Why it scores: Writes W(t) as the initial amount plus the accumulation integral of R, and evaluates W(6) = 40. Omitting the initial 4 gallons loses a point.
How to score points on AP Calculus FRQs
State the width and endpoints clearly in a Riemann sum, then show the sum.
Integrate a rate to get accumulated change; include units.
Accumulation functions start from an initial value plus an integral: W(t) = W(0) + ∫ R.
Use the FTC — find an antiderivative and evaluate at the bounds.
Answer the verb. "Approximate" wants the sum; "find" wants the exact value.