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← Calculus II
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Practice deliberately

Work through Calculus II unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Calculus II coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Calculus II syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use calculus practice questions with answers.

01

Integration techniques

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a integration techniques problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Applications of integration

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a applications of integration problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Improper integrals

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a improper integrals problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Sequences

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a sequences problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Infinite series

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a infinite series problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Power series

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a power series problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Parametric and polar curves

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a parametric and polar curves problem from its wording, carry out the governing method, and check that the result is reasonable.

Calculus II questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Evaluate the integral of x times e^x from 0 to 1.

Calculus II · Integration techniques

  1. 1
  2. e - 1
  3. e
  4. e + 1
Show the worked answer
  1. Use integration by parts with u = x and dv = e^x dx.
  2. Then du = dx and v = e^x, so the antiderivative is x e^x - e^x.
  3. At x = 1 that is e - e = 0.
  4. At x = 0 it is 0 - 1 = -1, so the value is 0 - (-1) = 1.

Answer: 1

Question 2: The region under y = the square root of x from x = 0 to x = 4 is revolved about the x-axis. What is the volume?

Calculus II · Applications of integration

  1. 8 pi
  2. 16 pi
  3. 4 pi
  4. 32 pi / 3
Show the worked answer
  1. Use disks: the volume is pi times the integral of (radius)^2, and the radius is the square root of x.
  2. The square of the square root of x is just x.
  3. The integral of x from 0 to 4 is x^2 / 2 = 16 / 2 = 8.
  4. Multiply by pi: the volume is 8 pi.

Answer: 8 pi

Question 3: Evaluate the improper integral of 1 / x^3 from 1 to infinity.

Calculus II · Improper integrals

  1. 1/2
  2. 1
  3. diverges
  4. 1/3
Show the worked answer
  1. An antiderivative of x^-3 is -1 / (2x^2).
  2. At the upper limit, -1 / (2x^2) approaches 0 as x grows.
  3. At x = 1 the antiderivative is -1/2.
  4. 0 - (-1/2) = 1/2, so the integral converges to 1/2.

Answer: 1/2

Question 4: What is the limit of the sequence a(n) = (1 + 2/n)^n as n approaches infinity?

Calculus II · Sequences

  1. e^2
  2. 1
  3. e
  4. infinity
Show the worked answer
  1. This matches the standard form (1 + k/n)^n, whose limit is e^k.
  2. Here k = 2.
  3. You can also take logs: n ln(1 + 2/n) is about n times 2/n = 2 for large n.
  4. So the limit is e^2, about 7.389.

Answer: e^2

Question 5: Find the sum of the series 5 + 5(2/3) + 5(2/3)^2 + ... .

Calculus II · Infinite series

  1. 15
  2. 7.5
  3. 10
  4. the series diverges
Show the worked answer
  1. This is geometric with first term 5 and ratio 2/3.
  2. Because the ratio is between -1 and 1, the series converges to first term / (1 - ratio).
  3. 1 - 2/3 = 1/3.
  4. 5 divided by 1/3 is 5 x 3 = 15.

Answer: 15

Question 6: Find the sum from n = 1 to infinity of 1 / (n(n + 1)).

Calculus II · Infinite series

  1. 1
  2. 1/2
  3. 2
  4. the series diverges
Show the worked answer
  1. Split the term into partial fractions: 1 / (n(n + 1)) = 1/n - 1/(n + 1).
  2. The partial sum is (1 - 1/2) + (1/2 - 1/3) + ... + (1/N - 1/(N + 1)).
  3. Almost everything cancels, leaving 1 - 1/(N + 1).
  4. As N grows, 1/(N + 1) goes to 0, so the sum is 1.

Answer: 1

Question 7: What is the radius of convergence of the series with terms n(x - 2)^n / 3^n?

Calculus II · Power series

  1. 3
  2. 1
  3. 1/3
  4. infinite
Show the worked answer
  1. Apply the ratio test to the size of consecutive terms.
  2. The ratio simplifies to ((n + 1) / n) times |x - 2| / 3.
  3. (n + 1) / n approaches 1, so the limit is |x - 2| / 3.
  4. Convergence needs that limit below 1, so |x - 2| < 3 and the radius is 3.

Answer: 3

Question 8: What is the area enclosed by the polar curve r = 4 sin(theta)?

Calculus II · Parametric and polar curves

  1. 4 pi
  2. 8 pi
  3. 16 pi
  4. 2 pi
Show the worked answer
  1. Polar area is one half the integral of r^2, and the full curve is traced from theta = 0 to pi.
  2. r^2 = 16 sin^2(theta), and the average value of sin^2 over that range is 1/2.
  3. So the integral of 16 sin^2 from 0 to pi is 16 x (pi / 2) = 8 pi.
  4. Half of 8 pi is 4 pi. This makes sense: the curve is a circle of radius 2.

Answer: 4 pi

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Calculus II questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Integration techniquesEvaluate the integral of x times e^x from 0 to 1.1
2Applications of integrationThe region under y = the square root of x from x = 0 to x = 4 is revolved about the x-axis. What is the volume?8 pi
3Improper integralsEvaluate the improper integral of 1 / x^3 from 1 to infinity.1/2
4SequencesWhat is the limit of the sequence a(n) = (1 + 2/n)^n as n approaches infinity?e^2
5Infinite seriesFind the sum of the series 5 + 5(2/3) + 5(2/3)^2 + ... .15
6Infinite seriesFind the sum from n = 1 to infinity of 1 / (n(n + 1)).1
7Power seriesWhat is the radius of convergence of the series with terms n(x - 2)^n / 3^n?3
8Parametric and polar curvesWhat is the area enclosed by the polar curve r = 4 sin(theta)?4 pi

How the generated Calculus II sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Calculus II practice page

What does this page cover?

It covers all 7 Calculus II units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.