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McGill University • Montreal

MATH 141 Tutor in Montreal, McGill Calculus 2

MATH 141 is examined without a calculator, and that changes what a good answer looks like. Every result is exact, a multiple of π, a logarithm, an arctangent, and the marks go to the choices you name along the way: the u and dv of an integration by parts, the substitution and its triangle, the convergence test and each of its hypotheses. Most calculus you find by searching ends on a decimal, or states a verdict without the check that earns it. Everything on this page is written for MATH 141 as it is actually examined.

24 chapters, 260 corrected exercises and 24 revision sheets, plus 2 practice papers. Free, no account needed: the full worked solution is on the page.

The definite integral

  1. 1 Riemann sums and the definite integral Ten exercises built on one idea: an integral is the limit of a sum of signed rectangles, so every answer starts by writing the width of a strip and its sample point. Part A is the mechanics: sigma notation and the sums of powers, left, right and midpoint sums on a decreasing curve, distance from a table of velocities, the integral computed as a limit with the sum formulas, and a limit of sums read back as an integral and evaluated by geometry. Part B works at exam level: signed area and the properties of the integral, a parabola on which neither sum stays on one side, five statements to correct, the energy of a solar array from its power readings, and a cubic whose integral is zero while its area is not. No antiderivative anywhere, and no calculator, as the course requires. Sheet
  2. 2 The Fundamental Theorem of Calculus Ten exercises built on one idea: an integral is a signed accumulation, and the Fundamental Theorem links it to its rate in both directions. Part A is the mechanics: differentiating an accumulation with the integrand read at the moving bound, two moving bounds and the logarithm defined as an integral, an accumulation function read off a graph, definite integrals rewritten by algebra before the table, and the cases where F(b) minus F(a) gives a wrong number. Part B works at final exam level: indefinite integrals checked by differentiating, displacement against distance, five statements to correct, a storage tank during a storm, and a five-part final exam question. No calculator anywhere, as the course requires. Sheet
  3. 3 The substitution rule Ten exercises built on one rule: a substitution is finished only when the integral contains no x at all, not in the integrand, not in the differential, not in the bounds. Part A is the mechanics: choosing the inside function, definite integrals with bounds carried through u (in the wrong order too), a stray x rewritten through u, logarithms of the denominator with their absolute value, and odd and even integrands on symmetric intervals. Part B works at final exam level: integrals that look alike but give a logarithm, an inverse tangent or an inverse sine, an unknown function known only through two integrals, five statements to correct, the charge through three circuits, and the reflection x to a minus x. No calculator anywhere, as the course requires. Sheet

Techniques of integration

  1. 4 Integration by parts Ten exercises built on one idea: integration by parts does not compute an integral, it trades it for another one, and the choice of u is judged by whether the new integral is easier. Part A is the mechanics: choosing u and dv and seeing the wrong choice fail, ln x, arctan x and arcsin x alone, repeated parts and the tabular method, cyclic integrals solved like an equation, and the bracket of a definite integral read as a rectangle. Part B works at final exam level: a reduction formula and how it amplifies an error, parts on a function known only by a few values, five statements to correct, a damped oscillation whose swings shrink by a factor e, and x arctan x with a clever constant in v. Exact answers only, no calculator, as the course requires. Sheet
  2. 5 Trigonometric integrals Ten exercises built on one idea: a trigonometric integral is a substitution in disguise, so you save one factor from the odd power, convert the rest with a Pythagorean identity, and lower the degree when every power is even. Part A is the mechanics: odd powers of sine and cosine, even powers and the half-angle formulas, tangent and secant, the integrals of sec and sec cubed, and products of different frequencies. Part B works at final exam level: substitutions that hide the trigonometric integral, symmetry and a square root that silently changes a sign, five statements to correct, the Mercator map, and the energy of a two-tone signal. No calculator anywhere, as the course requires. Sheet
  3. 6 Trigonometric substitution Ten exercises built on one idea: a trigonometric substitution is a round trip. The form under the root chooses the sine, the tangent or the secant, the interval of the angle fixes the sign of the root, and the answer comes back to x through the triangle or stays in the angle because the limits moved. Part A is the mechanics: the three forms, the circle behind the root of 9 minus x squared, a sum of squares with limits that move, the secant on both branches, and the area of an ellipse. Part B works at final exam level: completing the square, when a plain substitution is cheaper, five statements to correct, a fuel gauge for a horizontal tank, and a hyperbola whose sector has a logarithm for an angle. No calculator anywhere, as the course requires. Sheet
  4. 7 Partial fractions Ten exercises built on one rule: the shape of the answer is decided before any constant, in a fixed order, degree check and long division first, complete factoring second, one term per power third. Part A is the mechanics: the quotient the cover-up cannot see, distinct linear factors with their inner coefficient and absolute values, repeated factors that integrate to a power, irreducible quadratics that give a logarithm plus an arctangent, and the substitutions u equals root x and u equals e to the x. Part B works at final exam level: forms without constants, denominators to factor by grouping and by the rational root test, five statements to correct, the time course of a second-order reaction, and a complete final question with an exact answer. No calculator anywhere, as the course requires. Sheet
  5. 8 Strategy for integration Ten exercises on the naked integral, the one that arrives on an exam with no chapter title above it. Part A trains the first move: simplify before integrating, find the derivative hiding in the integrand, substitute then integrate by parts, rationalize then divide and split into partial fractions, and sort three families of lookalikes that call for three different tools. Part B works at final exam level: one integral solved by three routes of very different lengths, integrands with no elementary antiderivative and how to bound them or make them cancel, five statements to correct, a four-integral final exam question, and the charge carried by three electric currents. Every answer is exact, with no calculator, as the course requires. Sheet
  6. 9 Improper integrals Ten exercises built on one idea: an improper integral is a limit, one limit per problem point, and it converges only if every piece converges on its own. Part A is the mechanics: infinite bounds, integrands that blow up at an endpoint, the p-integrals and their two opposite rules, the asymptote hidden inside the interval, and the principal value that fakes a zero. Part B works at final exam level: the comparison test used in the right direction, integrals improper at both ends, five statements to correct, the area under a drug concentration curve, and the integrals of x to the n times e to the minus x. No calculator anywhere, as the course requires. Sheet

Applications of integration

  1. 10 Areas between curves and average value Ten exercises built on one idea: an area is a sum of slice lengths, and a length is top minus bottom, or right minus left, on each piece. Part A is the core gesture three times with a different trap each time, a parabola dipping below the axis, a cubic crossing a line three times, a sideways parabola sliced the wrong way, then the average value and the mean value theorem for integrals. Part B chooses the variable that avoids splitting, cuts a region in half with a line, corrects five statements, measures degree-hours on a daily temperature curve, and ends on a final exam question with sin x against sin 2x. Exact answers, no calculator, as the course requires. Sheet
  2. 11 Volumes: disks and washers Ten exercises on one idea: slice perpendicular to the axis, measure every radius from the axis, and square each radius before subtracting. Part A is the mechanics: disks about the x-axis with the square that picks the technique, washers about both coordinate axes where the outer curve changes, rotations about y = -1, y = 4 and x = 2, a logarithm region about four axes, and solids with known cross-sections on a disk base. Part B works at final exam level: the torus, a wedge sliced two ways, five statements to correct, a hemispherical basin filled to a depth h, and the region between cosine and sine. No calculator anywhere, as the course requires. Sheet
  3. 12 Volumes by cylindrical shells Ten exercises built on one gesture: draw the strip parallel to the axis, read its radius as a distance to the axis and its height as a length, and only then write the integral. Part A turns regions about the y-axis, the x-axis, three vertical lines, a horizontal line above a region that straddles the x-axis, and a region between two curves that dips below it. Part B makes the choice between shells and washers systematic, computes the same solid both ways, corrects five statements from real papers, prices the rubber of an O-ring, and ends on a final exam question with four axes. Every answer is exact, with no calculator, as the course requires. Sheet
  4. 13 Work Ten exercises built on one question: what varies, the force or the distance? Part A is the mechanics: work read as the area under a force graph, springs where the stretch is not the length, cables whose pieces rise their own depth, a cylindrical tank where the distance is not y, and a cone pumped vertex down and vertex up. Part B works at final exam level: a trough in foot-pounds without g, round tanks and the choice of origin, five statements to correct, a leaking bucket on a rope, and a payload lifted against gravity to infinity. Exact answers, no calculator. Sheet
  5. 14 Arc length and surface area Ten exercises built on one idea: length and surface area are measured along the curve, so every integral carries ds, the hypotenuse of dx and dy, never a bare dx. Part A is the mechanics: where the arc length formula comes from, the perfect squares that exam curves are built on, the secant and the hyperbolic cosine, switching to y when the slope blows up, and the arc length function. Part B works at final exam level: surfaces about the x-axis, the y-axis and a shifted line, five statements to correct, a chain hanging between two posts, and Gabriel's horn, finite volume and infinite area. No calculator anywhere, as the course requires. Sheet
  6. 15 Moments, centres of mass and hydrostatic force Ten exercises built on one idea: every quantity of the chapter is a sum of pieces times their arm, measured to the axis for a moment and from the surface for a pressure. Part A is the mechanics: point masses on a rod and in the plane, the centroid under one curve and its half-height strip, the region between two curves and the difference of squares, composite plates and a plate with a hole, and the theorem of Pappus run in both directions. Part B works at final exam level: triangular and inclined plates under water, a circular porthole where symmetry cancels an integral, five statements to correct, a trapezoidal dam drawn down in summer, and a parabolic canal gate with its centre of pressure. Exact answers, no calculator, as the course requires. Sheet
  7. 16 Probability density functions Ten exercises built on one idea: a probability is an area under the density, never its height. Part A is the mechanics: deciding whether a function is a density, finding the constant k, building the cumulative distribution function piece by piece, the mean as a balance point and the median as a half-area cut, and the exponential density of a waiting time. Part B works at final exam level: the normal density handled by symmetry alone, heavy tails whose mean is infinite, five statements to correct, two lifetime models for a pump under warranty, and a call centre whose waiting time has two phases. No calculator anywhere, as the course requires. Sheet

Sequences and series

  1. 17 Sequences Ten exercises on the first chapter of series in MATH 141, built on one idea: the limit of a sequence is decided by its tail, and every shortcut needs its permit first. Part A is the toolkit: reading a formula and a tail, L'Hôpital through a function of a real variable and the converse that fails, the squeeze theorem with bounds that share their limit, geometric sequences and the growth ladder up to factorials, and the limit (1 + x/n) to the n with the form 1 to the infinity. Part B works at final exam level: monotone and bounded sequences, a recursion solved in the right order, five statements to correct, compound interest climbing to continuous compounding, and Heron's square root with a guaranteed error. No calculator anywhere, as the course requires. Sheet
  2. 18 Series: geometric, telescoping and the divergence test Ten exercises built on one idea: a series is the limit of its partial sums, never of its terms. Part A is the mechanics: partial sums computed and read back, geometric series whose first term and ratio hide behind shifted indices and doubled exponents, telescoping series revealed by partial fractions, the divergence test and its silence, and the harmonic series proved divergent by grouping. Part B works at final exam level: repeating decimals as fractions, harder telescoping with factorials and logarithms, five statements to correct, repeated doses of a medication and their steady state, and the Koch snowflake. No calculator anywhere, as the course requires. Sheet
  3. 19 The integral test and the comparison tests Ten exercises built on one idea: a series is judged against something whose fate is already known, an integral or a benchmark series, and only after the hypotheses of the test are written down. Part A is the mechanics: the integral test with its three hypotheses, p-series in disguise, direct comparison pointing the right way, limit comparison with the dominant term, and the remainder estimate that turns three terms into two correct decimals. Part B works at final exam level: logarithms on the border p = 1, terms like sin(1/n) and 1 minus cos(1/n), five statements to correct, a leaning tower of books, and a family of series treated with every tool of the chapter. No calculator anywhere, as the course requires. Sheet
  4. 20 Alternating series Ten exercises built on one habit: separate the sign from the size. Part A is the mechanics: the alternating series test with both hypotheses written out, a decrease proved by a derivative when numerator and denominator both grow, the remainder bound and the sign of the error, the number of terms for a given accuracy, and the classification absolute, conditional or divergent. Part B works at final exam level: the alternating harmonic series summed to ln 2 through a Riemann sum, three series that break the test in three ways, five statements to correct, an autofocus motor whose position converges while its travel does not, and a one-parameter family that uses every verdict of the chapter. No calculator anywhere, as the course requires. Sheet
  5. 21 Ratio and root tests, and choosing a test Ten exercises built on one idea: the ratio and root tests compare a series with a geometric one and read only a limit L, below 1 converges, above 1 diverges, equal to 1 says nothing, so the test is chosen by the form of the general term. Part A is the mechanics: factorials simplified before the limit, n-th powers under the root test, the silent case L = 1, signed terms and recursive terms. Part B works at final exam level: a strategy drill, look-alike pairs that need different tests, five statements to correct, the expected number of retransmissions with a geometric tail bound, and a family where the ratio test falls silent at exactly one value. No calculator anywhere. Sheet
  6. 22 Power series Ten exercises built on one idea: the ratio test finds the radius, never the interval, and each endpoint is a separate series with its own test. Part A is the mechanics: the four shapes of an interval, a centre other than zero, missing powers, radii zero and infinite, series built from 1/(1 - u), partial sums against the function, and differentiation term by term. Part B works at final exam level: ln(1 + x) and arctan x by integration, endpoints lost and gained, five statements to correct, expected waiting times in a dice game, and a closed form that sums four numerical series. No calculator anywhere, as the course requires. Sheet
  7. 23 Taylor and Maclaurin series Ten exercises built on one formula read in both directions: the coefficient of the n-th power is the n-th derivative at the centre over n factorial. Part A is the mechanics: coefficients from the definition, the table of Maclaurin series and substitution into it, the binomial series, products, quotients and compositions, and derivatives read back from a series. Part B works at final exam level: limits by series, integrals with no elementary antiderivative, five statements to correct, goals in a hockey game summed exactly, and one function followed through every use of its series. No calculator anywhere, as the course requires. Sheet
  8. 24 Taylor polynomials and error bounds Ten exercises built on one idea: an approximation is a value and a guaranteed bound, never a value alone. Part A is the mechanics: building Taylor polynomials and reading them against the graph, choosing M at the worst point of the interval, approximating e to the 0.2 and the cube root of 1.1 by hand, the alternating estimate applied to a Taylor series, and the degree needed for a given accuracy. Part B works at final exam level: the interval on which a polynomial is good enough, cos 31 degrees centred at pi over 6, five statements to correct, the pendulum, the Lorentz factor and a GPS clock, and ln 2 to six decimals with six terms. No calculator anywhere, as the course requires. Sheet

When the chapters are done

Full papers under exam conditions, without a calculator. None of their questions repeats one already solved in the chapter sets above: the situations are new, so they measure what you can do rather than what you remember reading.

Practice exam

Practice midterm, MATH 141 at McGill

A two hour practice midterm for MATH 141, eight questions and one hundred points, on the first nine chapters of the course, with the full solution under every question. Part A asks four short questions: right sums and the definition of the integral, two moving bounds and a substitution whose bounds a classmate forgets, four integrals with no technique announced, and four improper integrals to settle. Part B asks four long problems: a reduction formula for powers of sine, three trigonometric substitutions, a quartic denominator to factor and decompose, and a comparison question with the infinity minus infinity trap. No calculator, every answer exact.

  • 8 questions
  • 100 points
  • 120 minutes
Sit the paper

Practice exam

Practice final exam, MATH 141 at McGill

A three hour practice final for MATH 141, twelve questions and one hundred points on the whole course, weighted like a real cumulative final, with the full solution under every question. Part A weighs the techniques: four integrals with no technique announced and four improper integrals, two to evaluate and two to decide. Part B weighs the applications: an area with its centroid and Pappus, one solid computed by shells and by washers, a paraboloid tank to pump out, an arc length with its surface of revolution, and a probability density. Part C weighs sequences and series: five convergence verdicts, two alternating series with an error bound, an interval of convergence with both endpoints, a Maclaurin series built from the table, and arctan 1.1 by Taylor's inequality. No calculator, every answer exact.

  • 12 questions
  • 100 points
  • 180 minutes
Sit the paper

What MATH 141 covers

The official description fits on one line: the definite integral, techniques of integration, applications, and an introduction to sequences and series. It is a four-credit course that follows MATH 139 or MATH 140, and it is closed to anyone holding the CEGEP integral calculus objective, 00UP. That restriction tells you who is in the room: students meeting integration techniques and infinite series for the first time.

What the line hides is that the course asks two different skills of you. The first half rewards computation: recognising which technique an integral calls for, carrying it through, and setting up an area, a volume or a quantity of work as the right integral before computing anything. The second half rewards justification: a series converges or diverges for a reason you have to write down, and a verdict without that reason earns almost nothing. Students who prepare the series chapters the way they prepared the integrals lose marks there, even when their answers are right.

The chapters above follow the order of the course and of its textbook, Stewart’s Calculus: Early Transcendentals, chapters 5 to 8 and 11. Each one has a corrected exercise set, with the full solution written out on the page, and a revision sheet that answers a different question: not what the chapter says, but what loses marks on it.

Where MATH 141 marks are actually lost

Forgetting the dx when the variable changes

A substitution changes three things at once: the integrand, the differential and, on a definite integral, the bounds. Students who replace only the first, or keep the old bounds after switching to u, lose the question at its first line. Write du next to dx before touching anything else.

Computing before drawing

Areas, volumes and work are marked on the set-up. A sketch shows which curve is on top, where the curves cross, how far the axis of rotation sits from the slice, and how far each layer of water has to travel. Most wrong answers in the applications half come from an integral that was never the right one.

Missing the hidden improper integral

An integrand that blows up inside the interval makes the integral improper even when both bounds are finite, and applying the Fundamental Theorem across the break produces a confident, wrong number. Check the integrand on the whole interval before evaluating anything.

Naming a test without checking it

The integral test needs a positive, continuous, decreasing function; the alternating series test needs terms that decrease to zero; a comparison needs the inequality in the direction that concludes. The divergence test can only ever prove divergence. On a series question, the hypotheses written on the page are the answer.

Frequently asked questions

What is MATH 141 at McGill?

MATH 141, Calculus 2, is McGill University’s four-credit second course in single-variable calculus. The official description is short: the definite integral, techniques of integration, applications, and an introduction to sequences and series. In practice that means Riemann sums and the Fundamental Theorem, substitution, integration by parts, trigonometric integrals and substitutions, partial fractions and improper integrals; areas, volumes, work, arc length, centres of mass and probability; then sequences, the convergence tests, power series and Taylor series.

I took Calculus 2 in CEGEP. Do I need MATH 141?

Usually not. MATH 141 is closed to students who have CEGEP objective 00UP or its equivalent, which is the CEGEP integral calculus course. The students in the room are therefore mostly first-year students from outside Quebec, international students and U0 students who took MATH 140 in the fall: most of them are meeting integration techniques and series for the first time.

Can I use a calculator on MATH 141 exams?

Assume you cannot: MATH 141 midterms and finals have typically been written without calculators, so check the rule on your own course outline. That is why every answer on this page is exact, written with pi, logarithms and arctangents rather than decimals, and why every solution names its choices. A method the marker can follow earns the marks even when an arithmetic slip spoils the final number.

Which part of MATH 141 is the hardest?

Series, for most students, and for a precise reason: the integration half of the course rewards computation, while a convergence question rewards a justification. Naming the right test is not enough; its hypotheses have to be checked on the page, one by one, and a correct verdict without them is worth very little. The series chapters below are written around that gesture.

Do you offer MATH 141 tutoring in Montreal?

Yes. I tutor MATH 141 and the other first-year McGill math courses in Montreal, in person and online. I am a McGill graduate and I work on your own course outline, WeBWorK assignments and past midterms rather than on a generic syllabus.

Other first-year courses

The same free corrected material exists for other courses, some of it in English and the rest in French.

COMP 202, Foundations of Programming (McGill)PHYS 101, Introductory Mechanics (McGill)MATH 133, Linear Algebra and Geometry (McGill)MATH 140, Calculus 1 (McGill)MATH 203, Calculus I (Concordia) Math tutoring in Montreal All revision sheets in English Higher education, in French
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