Worksheets · AP Calculus AB

AP Calculus AB Unit 8 Worksheet: Applications of Integration

20 exam-style multiple-choice questions on average value, motion, accumulation, area and volume, free to use in class. Good practice before the Unit 8 Progress Check in AP Classroom. Free response for this unit: Unit 8 FRQ set.

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

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AP Calculus AB Unit 8: Applications of Integration

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 1–8
  1. For 0≤x≤6, the region R lies under the graph of y=ex/2 and over the x-axis. What is the area of R?
    1. (A)e3−1
    2. (B)12e3−12
    3. (C)2e3
    4. (D)2e3−2
  2. For f(x)=1x, what is the average value of f over the interval 2≤x≤4?
    1. (A)38
    2. (B)332
    3. (C)ln⁡2
    4. (D)12ln⁡2
  3. A solid stands on the region enclosed by y=x2 and y=x, and its cross sections perpendicular to the y-axis are squares. Which integral expression equals the volume of the solid?
    1. (A)π∫02(2y−y2)2 dy
    2. (B)∫02(2y+y2)2 dy
    3. (C)∫02(2y−y2)2 dy
    4. (D)∫04(x−x2)2 dx
  4. −10123456789−2−101234y = v(t)
    The velocity v of an object moving on the x-axis is graphed above for 0≤t≤8; the graph is made of straight segments. The object starts at x(0)=−3. Where is it at t=8?
    1. (A)0
    2. (B)132
    3. (C)−192
    4. (D)72
  5. The cubic g(x)=−x3+4x2−2x and the line y=x enclose two regions. What is the total area of the two regions?
    1. (A)3712
    2. (B)83
    3. (C)94
    4. (D)56
  6. An object on the x-axis has acceleration a(t)=−4sin⁡(2t) for t≥0. It starts at x(0)=3 with velocity v(0)=9. Which of the following is its position function x(t)?
    1. (A)− ⁣sin⁡(2t)+11t+3
    2. (B)sin⁡(2t)+7t
    3. (C)sin⁡(2t)+7t+3
    4. (D)sin⁡(2t)+9t+3
  7. xy−3−2−1123−11234567891011Oy = 9y = 2x² + 1R
    In the figure above, R is the region between the parabola y=2x2+1 and the line y=9. Revolving R about the line y=9 produces a solid. What is its exact volume?
    1. (A)3712π15
    2. (B)2048π15
    3. (C)64π3
    4. (D)204815
  8. What is the area of the bounded region that lies between the curve x=y2 and the line x=4y−3?
    1. (A)43
    2. (B)83
    3. (C)263
    4. (D)10

Part B: A graphing calculator is allowed.

Questions 9–20
  1. A rectangular town is 8 miles long, measured along a lakeshore, and extends 3 miles back from the shore. The number of residents per square mile at a distance of x miles from the shore is f(x). Which of the following expressions gives the town's population?
    1. (A)∫08f(x) dx
    2. (B)8∫03f(x) dx
    3. (C)24∫03f(x) dx
    4. (D)∫03f(x) dx
  2. Revolving about the x-axis the region between the x-axis and the curve y=e−x2/4+1 from x=0 to x=2 produces a solid. To the nearest thousandth, what is its volume?
    1. (A)6.184
    2. (B)19.172
    3. (C)10.976
    4. (D)19.426
  3. −101234567−4−3−2−10123y = f(x)
    For 0≤x≤6, the graph of f shown above is made of straight segments and a semicircle. What is the average value of f over this interval?
    1. (A)−23
    2. (B)π12−13
    3. (C)π2−2
    4. (D)1+π12
  4. xy−3−2−1123−1123456Oy = g(x)y = f(x)R
    Let f(x)=5−x2 and g(x)=2x. The region R enclosed by the graphs of y=f(x) and y=g(x), shown in the figure above, is spun about the x-axis. To the nearest thousandth, what is the volume of the solid that results?
    1. (A)162.891
    2. (B)103.757
    3. (C)51.850
    4. (D)180.516
  5. Water flows into a reservoir from a river at the rate E(t)=10+8sin⁡ ⁣(πt8) and is released through a dam at the rate L(t)=7+2t, both in acre-feet per hour, where t is the number of hours since noon. At noon there are 50 acre-feet in the reservoir. What is the greatest amount of water in the reservoir for 0≤t≤8?
    1. (A)68.191 acre-feet
    2. (B)130.239 acre-feet
    3. (C)50.744 acre-feet
    4. (D)50.000 acre-feet
  6. Let f(x)=8x−x2 and g(x)=x2. What is the area of the region enclosed by the graphs of f and g?
    1. (A)21.554
    2. (B)11.314
    3. (C)9.133
    4. (D)31.794
  7. A conveyor loads grain into a silo at the rate g(t)=18 ⁣(1−e−0.3t) tons per hour, t hours after the conveyor is switched on. When the conveyor is switched on, the silo already contains 139 tons. How many tons of grain are in the silo 4 hours after the conveyor starts?
    1. (A)169.072 tons
    2. (B)189.314 tons
    3. (C)151.579 tons
    4. (D)30.072 tons
  8. A rowing boat drifts up and down a straight canal with velocity v(t)=1−2sin⁡ ⁣(t22) meters per second, 0≤t≤3. What is the distance between where the boat is at t=0.5 and where it is at t=3?
    1. (A)1.027
    2. (B)2.204
    3. (C)0.569
    4. (D)2.174
  9. During a storm, snow piles up on a deck at the rate S(t)=1.2+0.8sin⁡ ⁣(t23) centimeters per hour, where t is the number of hours since the storm began. By how many centimeters does the snow on the deck deepen between t=1 and t=4 hours?
    1. (A)0.549 centimeters
    2. (B)5.518 centimeters
    3. (C)4.230 centimeters
    4. (D)3.698 centimeters
  10. Let A(k) be the area between the graph of y=2x+1 and the x-axis on the interval [k,4]. For which k in (0,4) is A(k)=1.2?
    1. (A)2.509
    2. (B)0.822
    3. (C)0.667
    4. (D)1.744
  11. For 0≤t≤2.5 seconds, an object on a line moves with velocity v(t)=3e−t/2sin⁡(3t) meters per second. To the nearest thousandth, what total distance does it cover, in meters?
    1. (A)0.806
    2. (B)1.751
    3. (C)2.668
    4. (D)0.833
  12. Let f(x)=5cos⁡ ⁣(πx2) and g(x)=5(x−1)2. A solid has as its base the region R in the first quadrant bounded by the graphs of y=f(x) and y=g(x), and its cross sections perpendicular to the x-axis are isosceles right triangles, each with its hypotenuse in R. To the nearest thousandth, what is the volume of the solid?
    1. (A)1.386
    2. (B)0.693
    3. (C)1.875
    4. (D)0.379

    End of Version A

Answer key · Version A

12345678910
DDCDACBABD
8.48.18.78.28.68.28.108.58.38.9
11121314151617181920
BAABACCDCB
8.18.118.38.48.38.28.38.48.28.8

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 8?

About an eighth of the multiple choice, but more of the free response than any other unit: Unit 8 was 43 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (13.7%) and 28.1% of the free-response points on the 2012–2026 exams.

This worksheet has 20 questions in that mix: Part A (8) without a calculator and Part B (12) with one; the exam put 53% of its official Unit 8 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 8 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 8 (Applications of Integration), 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
Average value of a functionTopic 8.1112123
Average value from a graph or conceptTopic 8.11766
Position or velocity from an initial conditionTopic 8.2214126
Displacement and total distanceTopic 8.2212116
Amount from a rate in contextTopic 8.3211108
Rate in, rate out: maximum amountTopic 8.31111
Interpret or set up an integral in contextTopic 8.31331
Area between curves (dx)Topic 8.4112102
Area under one curveTopic 8.41652
Find k from a given areaTopic 8.41220
Area between curves (dy)Topic 8.51No official multiple-choice question is tagged to this topic in the 17 official exams (1969–2018); the course framework lists area with respect to y as its own topic (8.5), and a region bounded by curves written as x = h(y) comes up in free-response region questions.
Area when curves cross more than onceTopic 8.61111
Volume with square or rectangle cross sectionsTopic 8.71663
Volume with triangle or semicircle cross sectionsTopic 8.81221
Disc method about the x- or y-axisTopic 8.91662
Disc method about a line other than the axesTopic 8.101No official multiple-choice question is tagged to this topic in the 17 official exams (1969–2018); the course framework lists discs about a line other than an axis as its own topic (8.10), and revolving a region about a horizontal line is a regular free-response part.
Washer methodTopic 8.111221

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: disc or washer volume about a line other than the axes (FRQ); explain the meaning of a definite integral in context, with units (FRQ).

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

Why these kinds of question: AP Calculus AB Unit 8 Review: The Most-Tested Integration Application Questions.

Questions about this worksheet

Unit 8, Applications of Integration, has 12 AB topics in the College Board's course description: the average value of a function on an interval; position, velocity and acceleration with integrals, including displacement and total distance; accumulation functions and definite integrals in applied contexts; the area between curves as functions of x, as functions of y, and for curves that cross more than twice; volumes with square and rectangle cross sections, and with triangle and semicircle cross sections; and volumes of revolution by discs and washers, about the x- or y-axis and about other lines. Topic 8.13 (arc length) is BC only. The course description weights the unit at 10–15% of the multiple-choice section, and a Unit 8 question was on the free-response section of 14 of the 15 exams from 2012 to 2026.

Only in Part B, as on the exam. Questions 1–8 are Part A, with no calculator; questions 9–20 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 8 Part B questions need one: 10 of the 12. They are areas and volumes over regions whose curves meet where only a calculator finds them, such as y = 5 − x² and y = 2ˣ, amounts from rate models with units, total distance traveled, the greatest amount in a rate-in, rate-out model, and the k that splits a region in two. The others need none: an average value from a graph, from a given value or from an accumulation function, and which integral gives the population of a town. 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 8 questions into 19 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam’s multiple choice actually uses gets at least one question, and the most-tested get more (2 on position or velocity from an initial condition, 2 on displacement and total distance, 2 on amount from a rate in context). The 2 kinds that have never been the main topic of an official multiple-choice question are left out. Two of them are asked only on the free-response section: a volume of revolution about a line other than the axes (19 free-response questions since 1998) and the meaning of a definite integral in context, with units (25).

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 8 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.