Calculus 2: practice problems and exam questions

The second calculus course: integration techniques, sequences and series with convergence tests, power series and Taylor polynomials, and calculus in several variables with partial derivatives, gradients, critical points and double integrals. Depending on the curriculum, part of this material is taught as Calculus 3.

Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.

Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Integralspolynomials · exponentials, logs and substitution · areas between curvescalculations
  2. Sequences & seriesarithmetic and geometric series · convergence and power series · Taylor and Maclaurin polynomialscalculations
  3. Multivariable calculuspartial derivatives and double integrals · gradient, tangent plane and chain rule · critical points and integrals over regionscalculations
  4. Differential equationsexponential growth and decay · Newton's law of cooling · first-order linearcalculations

Exam tips for Calculus 2

  • Before any convergence test, check that the terms go to 0; if they do not, the series diverges.
  • Write the Taylor polynomial with the factorials in place before you substitute a number.
  • For a double integral, sketch the region first: the sketch gives the limits of the inner integral.

Worked examples

IntegralsEasy
xy3y = 0.5x

The region under y = 0.5x from x = 0 to x = 3 is rotated about the x-axis. Find the volume V of the solid (a cone).

y = 0.5x0 ≤ x ≤ 3

Find V

Show hints
  1. A slice at x is a disc of radius y = 0.5x.
  2. V = π∫₀3 (0.5x)² dx.
  3. = π·0.25·x³/3.
Show the answer and the step-by-step solution

Volumes of revolution: rotating the region under y = f(x) from x = a to x = b about the x-axis makes a solid whose cross-section at x is a disc of radius f(x). Adding the discs gives V = π∫ₐᵇ f(x)² dx. When the region lies between an outer curve R(x) and an inner curve r(x), each slice is a washer: V = π∫ₐᵇ (R(x)² − r(x)²) dx. Square the radii before you subtract them.

Disc method

V = π∫₀3 0.25x² dx

Integrate

V = π·0.25·3³/3 = 7.069

Answer

V = 7.069

Check: a cone of radius 1.5 and height 3 has (1/3)πr²h = 7.069.

Sequences & seriesMedium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 12/(k(k + 3)).

ak = 12/(k(k + 3))

Find S

Show hints
  1. Partial fractions: 1/(k(k + 3)) = (1/3)·(1/k − 1/(k + 3)).
  2. Write out the first few terms: most of them cancel (a telescoping sum).
  3. What remains is (12/3)·(1 + … + 1/3).
Show the answer and the step-by-step solution

Convergence: if the terms do not tend to 0 the series diverges (n-th term test). Σ1/kᵖ converges only for p > 1, and Σrᵏ only for |r| < 1. Compare a series with one of these when its terms behave like them for large k. Ratio test: L = lim |ak+1/ak|; root test: L = lim |ak|1/k. L < 1 converges, L > 1 diverges, L = 1 decides nothing. A power series Σck(x − x₀)ᵏ converges for |x − x₀| < R with R = lim |ck/ck+1|; an alternating series with terms decreasing to 0 converges.

Partial fractions

12/(k(k + 3)) = (12/3)·(1/k − 1/(k + 3))

Cancel

S = (12/3)·(1 + 1/2 + 1/3) = 7.333

Answer

S = 7.333

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Half-life and exponential decay calculatorIntegral calculator (definite and indefinite)

Related courses

Engineering Mathematics 2AP Calculus practice problemsEngineering MathematicsDifferential EquationsNumerical MethodsCalculus 1Introductory StatisticsStatistics 2: Statistical Inference
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.