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Integration and accumulation of change practice problems AP Calculus, Unit 6

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

Riemann sums and the trapezoidal rule, definite integrals, the fundamental theorem of calculus, and techniques of integration. 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 integration and accumulation of change

∫ab f(x) dx = F(b) − F(a)fundamental theorem, part 2
d/dx ∫ax f(t) dt = f(x)fundamental theorem, part 1
∫ xn dx = xn+1/(n + 1) + Cpower rule, n ≠ −1
∫ 1/x dx = ln|x| + Cthe reciprocal

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
Evaluate ∫02 3x2 dx.

Hint

Antidifferentiate, then substitute the limits.

Worked solution
  1. An antiderivative is x3.
  2. 23 − 0 = 8.
Q2Warm-upNo calculator
Use a left Riemann sum with 2 equal subintervals to estimate ∫02 x2 dx.

Hint

Width 1, heights from the left ends: x = 0 and x = 1.

Worked solution
  1. Each subinterval has width 1.
  2. Left heights: f(0) = 0 and f(1) = 1.
  3. Sum = 1 × (0 + 1) = 1, an underestimate since f is increasing.
Q3Warm-upNo calculator
d/dx ∫0x cos(t2) dt equals:

Hint

This is the fundamental theorem, part 1.

Worked solution
  1. The derivative of an accumulation function is the integrand at the upper limit.
  2. So the answer is cos(x2): B.

Exam-style questions

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

Q4Exam-styleNo calculator
Evaluate ∫1e (1/x) dx.

Hint

The antiderivative of 1/x is ln x.

Worked solution
  1. [ln x] from 1 to e = ln e − ln 1.
  2. = 1 − 0 = 1.
Q5Exam-styleNo calculator
Evaluate ∫01 2x(x2 + 1)3 dx.

Hint

Let u = x2 + 1. Change the limits too.

Worked solution
  1. u = x2 + 1, du = 2x dx; x = 0 gives u = 1, x = 1 gives u = 2.
  2. ∫12 u3 du = [u4/4] from 1 to 2.
  3. (16 − 1)/4 = 15/4 = 3.75.
Q6Exam-styleNo calculator
f(0) = 1, f(2) = 3 and f(4) = 7. Use the trapezoidal rule with these two subintervals to estimate ∫04 f(x) dx.

Hint

Average the two ends of each strip, times its width.

Worked solution
  1. Strip 1: 2 × (1 + 3)/2 = 4.
  2. Strip 2: 2 × (3 + 7)/2 = 10.
  3. Total = 14.
Q7Exam-styleNo calculator
F(x) = ∫2x2 √t dt. Find F'(2).

Hint

Fundamental theorem with the chain rule: f(upper limit) × derivative of the upper limit.

Worked solution
  1. F'(x) = √(x2) × 2x.
  2. F'(2) = 2 × 4 = 8.
Q8Exam-styleNo calculator
Find the average value of f(x) = x2 on [0, 3].

Hint

Average value = (1/(b − a)) ∫ab f(x) dx.

Worked solution
  1. ∫03 x2 dx = 9.
  2. Average = 9/3 = 3.

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 · 5 points · about 15 minutes

Water flows into a tank at a rate r(t) = 20 + 4t litres per minute, for 0 ≤ t ≤ 5, where t is in minutes. The tank holds 40 litres at t = 0.

  1. (a) How much water flows into the tank during the first 5 minutes?
  2. (b) Find the average rate of flow over the 5 minutes.
  3. (c) Water also leaks out at a constant 10 litres per minute. How much water is in the tank at t = 5?

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)

∫05 (20 + 4t) dt = 100 + 50 = 150 litres.

Part (b)

150/5 = 30 litres per minute.

Part (c)

40 + 150 − 50 = 140 litres.

Your mark: 0 of 5

Common mistakes in integration and accumulation of change

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