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Applications of integration practice problems AP Calculus, Unit 8

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Average value, displacement and distance, areas between curves, and volumes with cross-sections, discs and washers. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for applications of integration

Area = ∫ab (top − bottom) dxarea between curves
V = π∫ [R(x)]2 dxdisc method
V = π∫ ([R]2 − [r]2) dxwasher method
Distance = ∫ |v(t)| dttotal distance travelled

Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-upNo calculator
Find the average value of f(x) = 2x on [0, 4].

Hint

(1/(b − a)) × the integral.

Worked solution
  1. ∫04 2x dx = 16.
  2. 16/4 = 4.
Q2Warm-upNo calculator
Find the area between y = x and y = x2 for 0 ≤ x ≤ 1. You may give a fraction.

Hint

Which curve is on top between 0 and 1?

Worked solution
  1. On [0, 1], x ≥ x2.
  2. ∫01 (x − x2) dx = 1/2 − 1/3.
  3. = 1/6.
Q3Warm-upNo calculator
For a particle with velocity v(t), the total distance travelled from t = a to t = b is:

Hint

Distance counts movement in both directions as positive.

Worked solution
  1. ∫ v(t) dt is the displacement, where backward motion cancels forward motion.
  2. Distance needs |v(t)|. The answer is B.

Exam-style questions

Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.

Q4Exam-styleCalculator
The region under y = √x from x = 0 to x = 4 is rotated about the x-axis. Find the volume, as a decimal.

Hint

Disc method: V = π∫ y2 dx.

Worked solution
  1. V = π∫04 x dx.
  2. = π × 8 = 8π.
  3. ≈ 25.13.
Q5Exam-styleNo calculator
A particle has velocity v(t) = t − 2 for 0 ≤ t ≤ 4. Find the total distance it travels.

Hint

v changes sign at t = 2.

Worked solution
  1. From 0 to 2 it moves back: area 2.
  2. From 2 to 4 it moves forward: area 2.
  3. Total distance = 4 (the displacement is 0).
Q6Exam-styleNo calculator
Find the area enclosed by y = 4 − x2 and the x-axis. Give a decimal.

Hint

The curve crosses the x-axis at x = ±2.

Worked solution
  1. ∫−22 (4 − x2) dx.
  2. = [4x − x3/3] = (8 − 8/3) − (−8 + 8/3) = 32/3.
  3. ≈ 10.667.
Q7Exam-styleNo calculator
The base of a solid is the region under y = √x from x = 0 to x = 4. Cross-sections perpendicular to the x-axis are squares. Find the volume.

Hint

Each square has side √x, so area x.

Worked solution
  1. Area of a cross-section: (√x)2 = x.
  2. V = ∫04 x dx = 8.
Q8ChallengeCalculator
The region between y = x and y = x2 is rotated about the x-axis. Find the volume, to 4 decimal places.

Hint

Washer method: outer radius x, inner radius x2.

Worked solution
  1. V = π∫01 (x2 − x4) dx.
  2. = π(1/3 − 1/5) = 2π/15.
  3. ≈ 0.4189.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Calculator-active free-response style · 6 points · about 15 minutes

Let R be the region bounded by y = 2x and y = x2.

  1. (a) Find the area of R.
  2. (b) Find the volume when R is rotated about the x-axis.
  3. (c) R is the base of a solid with square cross-sections perpendicular to the x-axis. Find its volume.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

The curves meet at x = 0 and 2. Area = ∫02 (2x − x2) dx = 4 − 8/3 = 4/3.

Part (b)

π∫02 (4x2 − x4) dx = π(32/3 − 32/5) = 64π/15 ≈ 13.404.

Part (c)

∫02 (2x − x2)2 dx = 16/15 ≈ 1.067.

Your mark: 0 of 6

Common mistakes in applications of integration

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units