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Integration Accumulation Test – AP Calculus AB Math (Printable PDF)

Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

AP Calculus AB Math Tests – Mathbot, the Math Made Easy robot coach, with free printable tests for AP Calculus AB

This 45-minute test follows the AP Calculus AB style: two multiple-choice items and free-response questions with units and justification. Show every step. The total is 20 points, and no calculator is needed.

Standards: AP Calculus AB Unit 6 · Level: Grade 12 AP Calculus AB

Time: 45 minutes · 8 questions · Total: 20 points · Calculator: as allowed by your teacher

Question 1 Definite integral (2 points)

What is the value of ∫02 (3x2 − 2x) dx?

  1. 4
  2. 6
  3. 2
  4. 8

Question 2 Accumulation derivative (2 points)

Let H(x) = ∫0x √(t4 + 9) dt. What is H′(2)?

  1. 5
  2. 25
  3. 3
  4. 1/5

Question 3 Riemann sums from a table (3 points)

The rate r(t) at which a machine fills a tank, in gallons per minute, is given by the table.

t (min) 0 3 5 9
r(t) 8 10 7 2
  1. Find the left Riemann sum for ∫09 r(t) dt using the three subintervals. (1 point)
  2. Find the right Riemann sum using the same subintervals. (1 point)
  3. Find the trapezoidal sum and give the units. (1 point)

Question 4 Substitution (3 points)

Let f(x) = x2(x3 + 2)4.

  1. Find ∫ f(x) dx. (2 points)
  2. Evaluate ∫01 f(x) dx. (1 point)

Question 5 Accumulation function from a graph (3 points)

The graph of f consists of line segments. Let g(x) = ∫0x f(t) dt.

  1. Find g(2) and g(4). (1 point)
  2. Find g(6). (1 point)
  3. Find the absolute maximum value of g on [0, 6] and justify. (1 point)
123456−3−2−1123xy
Graph of f.

Question 6 Long division and completing the square (3 points)

Find each antiderivative.

  1. ∫ x2 − 3x + 5x − 1 dx (2 points)
  2. ∫ 1x2 + 4x + 5 dx (1 point)

Question 7 Properties of integrals (2 points)

For a function f, ∫14 f(x) dx = 7 and ∫16 f(x) dx = −2.

  1. Find ∫46 f(x) dx. (1 point)
  2. Find ∫41 (2f(x) + 3) dx. (1 point)

Question 8 Net change in a tank (2 points)

Water enters a tank at rate R(t) = 12 − 2t gallons per minute for 0 ≤ t ≤ 10 (a negative value means water is leaving). The tank holds 40 gallons at t = 0.

  1. How many gallons are in the tank at t = 6? (1 point)
  2. At what time on [0, 10] is the amount of water greatest? Justify. (1 point)
12345678910−8−44812xy
Graph of R(t).

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