Worksheets · AP Calculus AB

AP Calculus AB Unit 6 Worksheet: Integrals and Accumulation

20 exam-style multiple-choice questions on Riemann sums, accumulation functions, the Fundamental Theorem of Calculus and integration by substitution, free to use in class. Good practice before the Unit 6 Progress Check in AP Classroom. Free response for this unit: Unit 6 FRQ set.

Sigma Prep · sigmaprep.io/worksheets/ap-calculus-ab

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AP Calculus AB Unit 6: Integrals and Accumulation

Date Version A

Directions: Work each problem in the space below it, then choose the best of the four answer choices. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: No calculator is allowed.

Questions 1–15
  1. Suppose that G′(x)=4⋅3x. For some constant C, G(x) must equal which of the following?
    1. (A)4⋅3xln⁡3+C
    2. (B)4⋅3x+C
    3. (C)4⋅3x+1x+1+C
    4. (D)4⋅3xln⁡3+C
  2. The limit lim⁡n→∞∑k=1n5n ⁣(2+5kn)2 can be written as a definite integral. Which of the following does it equal?
    1. (A)∫25x2 dx
    2. (B)∫05x2 dx
    3. (C)∫05(2+x)2 dx
    4. (D)∫01(2+5x)2 dx
  3. Given that y=−6 at x=−2 and that y changes at the rate dydx=2x−4, find y as a function of x.
    1. (A)y=x2−4x
    2. (B)y=x2−4x−18
    3. (C)y=−8x−22
    4. (D)y=2x2−4x−22
  4. Pictured is the graph of f, pieced together from line segments, over −2≤x≤6. The definite integral ∫35f(x) dx equals which of the following?
    −3−2−101234567−2−101234y = f(x)
    1. (A)−23
    2. (B)−12
    3. (C)23
    4. (D)−43
  5. Let g(x)=∫2x(t2−3t) dt. What is g(1)?
    1. (A)−136
    2. (B)136
    3. (C)−76
    4. (D)−2
  6. Every antiderivative of w(x)=x(x2+2)2 can be written in which of the following forms?
    1. (A)16 ⁣(x2+2)3+C
    2. (B)13x(x2+2)3+C
    3. (C)12 ⁣(x2+2)2+C
    4. (D)13 ⁣(x2+2)3+C
  7. −2−10123456−101234y = f′(x)
    The derivative of a function f is graphed above for −1≤x≤5 (the graph of f′ is made of two line segments and a semicircle). Given f(1)=5, find f(−1).
    1. (A)5+π2
    2. (B)5−π
    3. (C)5−π2
    4. (D)−π2
  8. Approximate ∫17f(x) dx for f(x)=x2+3 by a right Riemann sum on three equal subintervals. What value results?
    1. (A)92
    2. (B)88
    3. (C)184
    4. (D)923
  9. Every antiderivative of p(x)=xx2−9 can be written in which of the following forms?
    1. (A)−12 ⁣(x2−9)+C
    2. (B)ln⁡∣x2−9∣+C
    3. (C)12ln⁡∣x2−9∣+C
    4. (D)2ln⁡∣x2−9∣+C
  10. A piecewise function is given by f(x)={−3for x<−2−2x+2for x≥−2. Find ∫−4−1f(x) dx.
    1. (A)−9
    2. (B)2
    3. (C)13
    4. (D)−1
  11. A student substitutes u=x+3 in ∫xx+3 dx and writes every factor in terms of u. Which of the following should the student get?
    1. (A)(u−3)∫u−1/2 du
    2. (B)∫u1/2 du
    3. (C)∫(u1/2+3u−1/2) du
    4. (D)∫(u1/2−3u−1/2) du
  12. For r(x)=ex+2, compute the exact value of ∫13r(x) dx.
    1. (A)4+e3−e
    2. (B)6+e3
    3. (C)e3−e
    4. (D)e−4−e3
  13. The function f is given by f(x)=e4+x4. Which of the following functions has derivative f(x) for all x?
    1. (A)∫0xe4+t4 dt
    2. (B)∫0x4e4+t dt
    3. (C)∫0xe4+x4 dt
    4. (D)e4+x44x3
  14. For the integrand w(x)=3xx2+144, find the exact value of ∫05w(x) dx.
    1. (A)35
    2. (B)3
    3. (C)32
    4. (D)6
  15. Let p(x)=x2x2+1. Writing C for the constant of integration, which choice is ∫p(x) dx?
    1. (A)x+arctan⁡x+C
    2. (B)x−arctan⁡x+C
    3. (C)x−arcsin⁡x+C
    4. (D)x+C

Part B: A graphing calculator is allowed.

Questions 16–20
  1. Bikes are returned to a bike-share station at a rate modeled by K(t)=6ln⁡ ⁣(t2+0.4)+6 bikes per hour, where t is measured in hours after the station opens. What is the number of bikes returned to the station between t=1 and t=5?
    1. (A)101.634 bikes
    2. (B)78.009 bikes
    3. (C)74.349 bikes
    4. (D)25.408 bikes
  2. A continuous function f is odd, and ∫02f(x) dx=−2. Find ∫−22(f(x)+sin⁡ ⁣(x2)) dx.
    1. (A)1.610
    2. (B)−1.195
    3. (C)−2.390
    4. (D)0.805
  3. Wood chips are blown into a storage bin at a rate of B(t) cubic meters per hour, with t in hours. The table below lists B(t) at several times.
    t (hours) 024610B(t) (cubic meters per hour) 202391720A trapezoidal sum with the four subintervals marked off by the times in the table is used to estimate ∫010B(t) dt. Based on that estimate, what is the average value of B(t), in cubic meters per hour, for 0≤t≤10?
    1. (A)175
    2. (B)17.2
    3. (C)43.75
    4. (D)17.5
  4. It is known that ∫03f(x) dx=−5 for a continuous function f. Find ∫03(4f(x)+e−x2/2) dx.
    1. (A)−3.750
    2. (B)−15.000
    3. (C)−21.250
    4. (D)−18.750
  5. With 4 subintervals of equal width, a trapezoidal sum gives a value less than ∫04f(x) dx, and a left Riemann sum gives a value greater than it. Which of the four graphs below could show y=f(x)?
    (A)
    −1012345−1012345y = f(x)
    (B)
    −1012345−1012345y = f(x)
    (C)
    −1012345−1012345y = f(x)
    (D)
    −1012345−1012345y = f(x)
    1. (A)Graph (A)
    2. (B)Graph (B)
    3. (C)Graph (C)
    4. (D)Graph (D)

    End of Version A

Answer key · Version A

12345678910
DCBABACCCD
6.86.36.86.66.56.96.76.26.96.6
11121314151617181920
DAABBCADDB
6.96.76.46.96.106.16.66.26.66.2

The second row is the correct letter; the third is the course topic each question tests.

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

What does the AP Calculus AB exam ask on Unit 6?

About a fifth of the multiple choice: Unit 6 was 61 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (19.4%), and it has more official multiple-choice questions across all 17 exams (159) than any other unit.

This worksheet has 20 questions in that mix: Part A (15) without a calculator and Part B (5) with one; the exam put 26% of its official Unit 6 multiple choice in the calculator section.

Counted from the official AP Calculus AB exams: multiple choice from 17 exams, 1969–2018, free response 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 6 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 6 (Integration and Accumulation of Change), and how many are on this worksheet
Question typeOn this worksheetOfficial questions, 1969–2018Exams it was on (of 17)On the 2012–2018 practice exams
Accumulation as area under a rate graphTopic 6.11221
Riemann or trapezoidal sum from a tableTopic 6.21885
Riemann or trapezoidal sum from a function or graphTopic 6.21331
Over- or underestimate for Riemann and trapezoidal sumsTopic 6.21441
Riemann sum notation to integralTopic 6.31221
FTC: derivative of an accumulation functionTopic 6.4111105
Analyze g(x) = integral of fTopic 6.51975
Definite integrals from graph geometryTopic 6.61764
Combine given integral valuesTopic 6.6110105
Integrals of piecewise and absolute-value functionsTopic 6.61762
Integral properties: symmetry, shifts and boundsTopic 6.61851
Find f(b) from f(a) and the integral of f′Topic 6.711286
Evaluate a definite integral with the FTCTopic 6.711783
Basic antiderivativesTopic 6.811183
Antiderivative with an initial conditionTopic 6.81332
u-substitution (indefinite)Topic 6.9116114
Integrals that give lnTopic 6.91873
Rewrite an integral in terms of uTopic 6.91866
u-substitution (definite)Topic 6.911082
Long division and completing the squareTopic 6.101331

Counts are of official multiple-choice questions whose main topic is that kind, on 17 College Board exams 1969–2018 (7 released 1969–1998, 2003 and 2008 released, the 2008 practice exam and the seven 2012–2018 practice exams). No official multiple-choice questions after 2018 are public. Left out: choose a technique for an antiderivative.

Free-response practice for this unit: 4 FRQs with a scoring guide.

Why these kinds of question: AP Calculus AB Unit 6 Review: The Most-Tested Riemann Sum and FTC Questions.

Questions about this worksheet

Unit 6, Integration and Accumulation of Change, has 11 AB topics in the College Board's course description: accumulation as the area under a rate graph; left, right, midpoint and trapezoidal sums; Riemann sums, summation notation and the definite integral; the Fundamental Theorem of Calculus and accumulation functions; the behavior of a function defined by an integral; the properties of definite integrals; evaluating definite integrals with antiderivatives; basic antiderivatives and indefinite integrals; integration by substitution; long division and completing the square; and choosing a technique for an antiderivative. Topics 6.11 to 6.13 (integration by parts, partial fractions and improper integrals) are BC only. The course description weights the unit at 15–20% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–15 are Part A, with no calculator; questions 16–20 are Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). Most of these Unit 6 Part B questions do not need one: a Riemann or trapezoidal sum from a table, over- and underestimates, the properties of definite integrals and combining given values of integrals. One in each version does: an amount from a rate model such as S(t) = 8 ln(t² + 0.4) + 6, found with the calculator's integral. A numerical answer is given to three decimal places, as on the exam. From May 2027 the exam's multiple-choice section is 29 questions without a calculator and 13 with one.

We tagged 760 official multiple-choice questions from 17 College Board exams (1969–2018) and the free-response questions from 1969 to 2026, sorted the Unit 6 questions into 21 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam actually uses gets one question. The one kind that has never been the main topic of an official multiple-choice question is left out.

The progress check is College Board's own end-of-unit quiz in AP Classroom, assigned by your teacher. Its questions are not public, and we do not have them: Sigma Prep is not affiliated with College Board. This worksheet is exam-style practice on the same unit: 20 original multiple-choice questions, weighted by how often the official exams ask each kind of question, with a no-calculator Part A and a calculator Part B. Time yourself, check the answer key, then redo the kinds you missed. For the free-response part, use the Unit 6 FRQ set, which has a scoring guide.

No. Every question here is original, written by Sigma Prep in the style of the exam. We studied which kinds of questions the official exams ask and how often; no official question is copied or reworded. Each answer was checked by an independent computer-algebra check before the sheet was published.

Yes. It is shown with the sheet, and the PDF prints it on its own last page. There is also a PDF without the key to hand out. Worked solutions are not included.

AP is a registered trademark of the College Board, which was not involved in making this worksheet and does not endorse it. All questions are original to Sigma Prep. Frozen 2026-10-24.