• 64 questions

AP Calculus Unit 8 Practice Test

Unofficial practice for Unit 8 of AP Calculus AB and BC: average value, motion, area, volume and arc length.

What is the average value of f(x) = 3x² + 1 on the interval [0, 2]?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 14, 2026.

Next step AP Calculus integrals: Unit 6AP Calculus Unit 7: differential equations

All 64 questions

Topic 8.1: Average value of a function 7

  • A room's temperature in degrees is T(t) = 60 + 12 sin(πt/12) for 0 ≤ t ≤ 12 hours. What is the average temperature over those 12 hours? Wikipedia, "Mean of a function"; course topic 8.1
  • The average value of a continuous function f on [2, 6] is 5. What is the integral of f from 2 to 6? Wikipedia, "Mean of a function"; course topic 8.1
  • For f(x) = x² on [1, 3] the average rate of change is 4. What is the average value of f on the same interval? Wikipedia, "Mean of a function"; course topic 8.1
  • What is the average value of f(x) = 1/x on [1, e]? Wikipedia, "Mean of a function"; course topic 8.1
  • What is the average value of f(x) = 3x² + 1 on the interval [0, 2]? Wikipedia, "Mean of a function"; course topic 8.1
  • What is the average value of f(x) = cos x on [0, π/2]? Wikipedia, "Mean of a function"; course topic 8.1
  • What is the average value of f(x) = √x on [0, 9]? Wikipedia, "Mean of a function"; course topic 8.1
Show all 64 questions (57 more)

Topic 8.2: Position, velocity and acceleration 8

  • A ball is thrown straight up with velocity v(t) = 20 − 10t meters per second. How far does it rise before it stops for an instant? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A cart moves at 5 meters per second for the first 4 seconds, then at −3 meters per second for the next 2 seconds. What is its displacement over the 6 seconds? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A particle has acceleration a(t) = 4t + 1 and velocity 3 at t = 0. What is its velocity at t = 2? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A particle has velocity v(t) = 4 − 2t and is at position x = 5 when t = 0. Where is it at t = 3? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A particle moves along a line with velocity v(t) = 3t² − 12 meters per second. What is its displacement from t = 0 to t = 3? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A particle moves along a line with velocity v(t) = 3t² − 12 meters per second. What total distance does it travel from t = 0 to t = 3? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • A particle moves along a line with velocity v(t) = sin t. What total distance does it travel from t = 0 to t = 2π? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1
  • For a particle moving along a line, what does the definite integral of its speed over a time interval give? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.2, essential knowledge CHA-4.C.1

Topic 8.3: Accumulation and net change 7

  • A population grows at a rate of P′(t) = 200e^(0.1t) people per year. By how many people does it grow from t = 0 to t = 10? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.E.1
  • A pump moves water at r(t) gallons per minute, with t in minutes. In what units is the integral of r(t) from 0 to 10? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.D.2
  • A tank holds 50 liters at t = 0. Water enters at 8 liters per minute and leaves at 2t liters per minute. How much is in the tank at t = 5? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.E.1
  • Cars pass a toll gate at a rate of R(t) cars per hour, with t in hours after midnight. What does the integral of R(t) from 9 to 12 give? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.D.2
  • Sand is added to a pile at 10 tons per hour and removed at 4 + 2t tons per hour. What is the net change in the pile over the first 3 hours? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.E.1
  • The volume of a reservoir changes at V′(t) = t − 4 thousand cubic meters per day for 0 ≤ t ≤ 8. What is the net change in volume over the 8 days? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.D.2
  • Water flows into a tank at r(t) = 6t − t² liters per minute for 0 ≤ t ≤ 6. How much water enters in those 6 minutes? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.3, essential knowledge CHA-4.D.2

Topic 8.4: Area between curves, functions of x 6

  • What is the area between y = cos x and y = sin x from x = 0 to x = π/4? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1
  • What is the area between y = e^x and the line y = 1 from x = 0 to x = 1? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1
  • What is the area of the region enclosed by y = 6x − x² and y = x²? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1
  • What is the area of the region enclosed by y = x² − 4 and y = 2 − x? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1
  • What is the area of the region enclosed by y = √x and y = x²? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1
  • Which integral gives the area between y = x² + 1 and y = x from x = 0 to x = 2? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.4, essential knowledge CHA-5.A.1

Topic 8.5: Area between curves, functions of y 4

  • A region lies between x = h(y) on the left and x = g(y) on the right for c ≤ y ≤ d. Which integral gives its area? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.5, essential knowledge CHA-5.A.2
  • What is the area of the region enclosed by x = y² and the line x = 4? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.5, essential knowledge CHA-5.A.2
  • What is the area of the region enclosed by x = y² and x = y + 2? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.5, essential knowledge CHA-5.A.2
  • What is the area of the region enclosed by x = y² − 1 and x = 1 − y²? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.5, essential knowledge CHA-5.A.2

Topic 8.6: Curves that cross more than twice 4

  • Two curves y = f(x) and y = g(x) cross several times between x = a and x = b. Which expression gives the total area between them? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.6, essential knowledge CHA-5.A.3
  • What is the area between y = cos x and the x-axis from x = 0 to x = π? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.6, essential knowledge CHA-5.A.3
  • What is the total area enclosed between y = x³ and y = x? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.6, essential knowledge CHA-5.A.3
  • What is the total area enclosed between y = x³ − 4x and the x-axis? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.6, essential knowledge CHA-5.A.3

Topics 8.7 and 8.8: Volumes with known cross sections 8

  • A cross section of a solid is a semicircle whose diameter s lies in the base. What is the area of the cross section? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.8, essential knowledge CHA-5.B.3
  • A solid lies between x = 0 and x = 5. Its cross section at x has area A(x) = 3x² square units. What is its volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.8, essential knowledge CHA-5.B.3
  • The base of a solid is the disk x² + y² ≤ 4. Cross sections perpendicular to the x-axis are squares. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.7, essential knowledge CHA-5.B.1
  • The base of a solid is the region between y = 4 − x² and the x-axis. Cross sections perpendicular to the x-axis are semicircles with diameter in the base. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.8, essential knowledge CHA-5.B.3
  • The base of a solid is the region under y = 2x for 0 ≤ x ≤ 3. Cross sections perpendicular to the x-axis are isosceles right triangles with one leg in the base. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.8, essential knowledge CHA-5.B.2
  • The base of a solid is the region under y = x for 0 ≤ x ≤ 2. Cross sections perpendicular to the x-axis are equilateral triangles. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.8, essential knowledge CHA-5.B.2
  • The base of a solid is the region under y = x for 0 ≤ x ≤ 3. Cross sections perpendicular to the x-axis are rectangles of height 2. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.7, essential knowledge CHA-5.B.1
  • The base of a solid is the region under y = √x for 0 ≤ x ≤ 4. Cross sections perpendicular to the x-axis are squares. What is the volume? College Board, AP Calculus AB and BC Course and Exam Description, topic 8.7, essential knowledge CHA-5.B.1

Topics 8.9 and 8.10: Disc method 8

  • The first-quadrant region bounded by y = x², y = 4 and the y-axis is revolved about the y-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.9
  • The region between y = x² and the line y = 4 is revolved about the line y = 4. What is the volume? Wikipedia, "Disc integration"; course topic 8.10
  • The region between y = √x and the line y = 3, from x = 0 to x = 9, is revolved about the line y = 3. Which integral gives the volume? Wikipedia, "Disc integration"; course topic 8.10
  • The region bounded by y = √x, the x-axis and the line x = 4 is revolved about the line x = 4. What is the volume? Wikipedia, "Disc integration"; course topic 8.10
  • The region under the line y = x/2 from x = 0 to x = 6 is revolved about the x-axis, making a cone. What is its volume? Wikipedia, "Disc integration"; course topic 8.9
  • The region under y = e^(x/2) from x = 0 to x = 2 is revolved about the x-axis. What is the volume of the solid? Wikipedia, "Disc integration"; course topic 8.9
  • The region under y = sin x from x = 0 to x = π is revolved about the x-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.9
  • The region under y = x² from x = 0 to x = 2 is revolved about the x-axis. What is the volume of the solid? Wikipedia, "Disc integration"; course topic 8.9

Topics 8.11 and 8.12: Washer method 7

  • A washer has outer radius R and inner radius r. What is its area? Wikipedia, "Disc integration"; course topic 8.11
  • The first-quadrant region between y = x² and the line y = 4 is revolved about the x-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.11
  • The region between y = 2x and y = x² is revolved about the x-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.11
  • The region between y = x and y = x² is revolved about the line y = 2. What is the outer radius of a washer at x? Wikipedia, "Disc integration"; course topic 8.12
  • The region between y = x and y = x² is revolved about the line y = −1. What is the volume? Wikipedia, "Disc integration"; course topic 8.12
  • The region between y = x and y = x² is revolved about the y-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.11
  • The region between y = √x and y = x/2 is revolved about the x-axis. What is the volume? Wikipedia, "Disc integration"; course topic 8.11

Topic 8.13: Arc length (BC only) 5

  • What is the length of the curve y = (1/3)(x² + 2)^(3/2) from x = 0 to x = 3? Wikipedia, "Arc length"; course topic 8.13
  • What is the length of the curve y = ln(cos x) from x = 0 to x = π/4? Wikipedia, "Arc length"; course topic 8.13
  • What is the length of the line y = 2x + 1 from x = 0 to x = 3? Wikipedia, "Arc length"; course topic 8.13
  • Which integral gives the length of the curve y = f(x) from x = a to x = b, when f′ is continuous? Wikipedia, "Arc length"; course topic 8.13
  • Which integrand gives the length of the curve y = x² from x = 0 to x = 1? Wikipedia, "Arc length"; course topic 8.13

How this list was compiled: Our own questions, written to Unit 8 (Applications of Integration; topics 8.1 to 8.13) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-14. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-14. The rule behind each answer was read on 2026-10-14 in the essential knowledge statement of the course description or the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus Unit 8 practice test has 64 questions of our own on Applications of Integration. It is unofficial and is not affiliated with or endorsed by the College Board, which owns AP.

Read the full guide

What Unit 8 covers

The College Board's course description for AP Calculus AB and BC, read October 14, 2026, gives Unit 8 thirteen topics. The sets here follow them: average value (7 questions), position, velocity and acceleration (8), accumulation and net change (7), area between curves (14 over topics 8.4 to 8.6), volumes with known cross sections (8), the disc method (8), the washer method (7) and arc length (5), which is BC only. The description gives topics no weights, so the spread is ours.

Worked answers

Every answer is worked from the curves, rates or velocities that the question prints, and each computed one was worked out a second time with computer algebra. Where one wrong choice is the usual slip, the explanation says which.

  • Average value divides the integral by b − a. Average rate of change divides f(b) − f(a) by b − a.
  • For total distance, find where the velocity changes sign and add the pieces as positive numbers.
  • In the washer method, square each radius first and then subtract. Squaring the difference is the usual slip.

Questions and answers

What is Unit 8 in AP Calculus?

Applications of Integration. Its thirteen topics run from the average value of a function (8.1) through motion, accumulation, area between curves and volumes to arc length (8.13).

How much of the AP Calculus exam is Unit 8?

From 10 to 15% of the multiple-choice section on the AB exam and from 5 to 10% on the BC exam, by the College Board's course pages.

Is Unit 8 the same in AP Calculus AB and BC?

Topics 8.1 to 8.12 belong to both courses. Topic 8.13, arc length and distance traveled, is BC only. The course description suggests 19 to 20 class periods for the unit in AB and 13 to 14 in BC.

Is this the AP Calculus Unit 8 progress check?

No. The College Board's Progress Check for Unit 8, about 30 multiple-choice and 3 free-response questions by the course description, is assigned by teachers in AP Classroom. The questions here were written for this page.

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