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CEGEP • Montreal

Integral Calculus, 201-SN3-RE (formerly 201-NYB)

Integral Calculus, 201-SN3-RE in the revised program (formerly 201-NYB), asks for two very different things: technical fluency with the methods of integration, then a new kind of rigour about series and their convergence. Many students get through the first part and stumble on the second.

19 fully corrected exercise sets (314 exercises), 19 revision sheets, and a practice paper. Free, no account needed. The same course in French.

Antiderivatives and integration techniques

  1. 1 Antiderivatives and the indefinite integral Two calculation exercises first, antiderivatives to compute directly, twelve with the table and ten by substitution; then the very first chapter of integral calculus: what an antiderivative really is, why the constant of integration is not a detail, the table of basic formulas with its gap at n = -1, linearity and the two false rules everyone invents, then substitution, from the direct reflex to substitutions that require back-substitution. Also covered: antiderivatives with an absolute value and on a domain in two pieces (where the constant can differ from one side to the other), the three forms to recognize at a glance (f'/f, f'f^n, f'e^f), antiderivatives that do not exist in elementary terms with the two fallback routes, and the method of undetermined coefficients, which replaces two integrations by parts with a linear system. A Part C adds five classics: bases other than e, a function recovered from its tangent line, five antiderivatives from a test paper, the deflection of a beam and the sales of a game. Sheet
  2. 2 Partial fraction decomposition Polynomial long division set out step by step, including a divisor that is not monic, which puts fractions in the quotient; reading a denominator to decide whether a fraction is already decomposed, with the discriminant as evidence; two simple poles handled by equating coefficients, with the graph and its two vertical asymptotes; the cover-up method, which gives each coefficient in one line instead of a system; the repeated factor and the proof that a single term cannot be enough; the irreducible quadratic factor and its linear numerator, justified by counting unknowns; the form of the decomposition given without computing the coefficients, on five denominators, two of which hide a quadratic that factors; five statements to correct, from the Sophie Germain identity to the real reach of the cover-up method; antiderivatives and an area, where the logarithm, the arctangent and the preliminary division share the work; and an epidemic among 1,200 students modelled by the logistic equation, which can only be integrated through partial fractions. Then a Part C of pure practice, in the format of the worksheets handed out in class: twenty-five decompositions to carry out, sorted by the shape of the denominator, from degree one to the fourth power, with two simple poles, the non-monic factor, the repeated factor, the irreducible quadratic alone and then squared, the cancellation that removes a pole, the difference of squares, the change of unknown, the difference of cubes, the Sophie Germain identity, and six forms to give without computing a single coefficient. Sheet
  3. 3 Integration techniques Two calculation exercises first: twenty integrals to compute directly, ten by parts and ten trigonometric powers or products; then substitution, from the reflex of spotting u and its differential to transformed limits of integration, integration by parts in its simple, repeated and cyclic forms, partial fractions with long division first and a repeated factor, powers of sine and cosine with their parity rules, trigonometric substitutions tied to their triangles, and the strategy exercise: choosing the right technique in thirty seconds. Also covered: powers of tangent and secant with their own two rules and the cyclic case of secant cubed, products of sines and cosines of different angles turned into sums, including the orthogonality that underlies Fourier series, combined techniques (substitute to clean up, integrate by parts to finish), and completing the square, which decides between logarithm, arctangent and arcsine according to the discriminant. A Part C adds five classics: the arches of x sin x, two routes to the same integral, an integral with three poles, the damped oscillator and the flood hydrograph of a river. Sheet
  4. 4 Substitution, parts and partial fractions: 20 exercises The three major integration techniques, covered in depth: the five forms where the derivative is already there, the substitution that leaves an x standing, rationalizing substitutions and the trap of the limits of integration; LIATE and why it works, the tabular method, the cyclic case, the boundary term and the invisible factor 1; then the four cases of partial fraction decomposition, the cover-up method and its limits, long division first, the irreducible quadratic and the real uses of the decomposition. Part D combines everything: look-alike pairs, substitution then parts, rational functions in disguise, the three free checks and twelve integrals to diagnose. Sheet
  5. 5 Trigonometric integrals: 20 exercises The longest set on the site, devoted entirely to trigonometric integrals and their inverses: the three cases of powers of sine and cosine, the secant and cosecant whose antiderivatives cannot be guessed, the tangent-secant family and its mirror image the cotangent-cosecant family with its signs, the three trigonometric substitutions and the trap of the two branches of the secant, completing the square, then arcsin, arccos, arctan, arccot and arcsec, integration by parts with the invisible factor 1, the absolute value in the inverse secant, improper integrals that bring out pi, and a final decision tree on eight mixed integrals. Sheet
  6. 6 Integration tricks: Weierstrass, King's rule, Wallis The direct sequel to integration techniques, for the integrals that resist the four methods of the course: King's rule, which folds the bounds onto each other, the Weierstrass substitution, which rationalizes sine and cosine, hyperbolic functions in place of trigonometric substitution, reduction formulas and Wallis integrals, the system of two integrals, the substitution that swaps x and its reciprocal, and the five pitfalls that cost marks to students who otherwise know how to integrate. A Part C adds five classics: Bioche's rules, integrating an inverse function without a formula, three famous integrals, Buffon's needle and the mean distance on a circle. Sheet

Indeterminate forms and improper integrals

  1. 7 L'Hôpital's rule and indeterminate forms, 201-SN3-RE (201-NYB) One calculation exercise first: twelve limits to compute directly with L'Hôpital's rule, from the trigonometric quotient to the exponential forms; then the seven indeterminate forms and what “indeterminate” really means, L'Hôpital's rule with its hypotheses and its two classic pitfalls, the product and difference forms brought back to a quotient, the three exponential forms handled through the logarithm, and the limits where L'Hôpital's rule goes in circles or fails outright. Added to these: the proof of the rule through Cauchy's Mean Value Theorem, with the circular argument hidden behind the limit of the sine, indeterminate forms at work (an oblique asymptote and its sign trap, a removable discontinuity, a parameter to adjust so that a limit is finite), the discussion of a limit according to a real parameter, including a quotient that grows to ten to the twenty-first power before tending to zero, and choosing the method in thirty seconds between algebra, the logarithm and L'Hôpital's rule. Sheet
  2. 8 Definite and improper integrals, 201-SN3-RE (201-NYB) One calculation exercise first: ten improper integrals to evaluate directly or to show divergent, from an infinite limit of integration to a singularity inside the interval; then Riemann sums worked by hand all the way to the exact limit, the Fundamental Theorem of Calculus in both directions, the techniques (substitution, integration by parts) and the properties that spare you a computation, the area between two curves and the volume of revolution, then improper integrals: the p-test at both ends, Gabriel's horn with its finite volume and infinite surface area, and the Comparison Test at exam level. Also covered: the properties that bound an integral without computing it (additivity over intervals, orientation of the limits, the absolute value inequality and the Mean Value Theorem for Integrals with its discontinuous counterexample), reading the accumulated area to deduce the intervals of increase, the maximum and the inflection points of an antiderivative from the graph of the function alone, a perfectly valid probability distribution that has no finite mean (power densities, moments and the Gamma function), and doubly improper integrals, where symmetry lies until the convergence of each half has been established. A Part C adds five classics: the Fundamental Theorem at work, the Laplace transform, the Limit Comparison Test, the perpetuity and the discharge of a capacitor. Sheet

Applications of the integral

  1. 9 Area between curves, 201-SN3-RE (201-NYB) One calculation exercise first: eight areas between two curves to compute directly, from a parabola against a line to a cubic that crosses a line three times; then the difference between an area and a signed integral, on a cubic whose integral is zero while its area is eight. The bounds that no problem gives and that you find by solving for the intersection, the test point that decides which curve is on top, the split you cannot skip when the roles switch, the horizontal strips that turn three integrals into one, sine against cosine over a full turn, a region bounded by three lines checked with the determinant, the parameter that cuts an area into two equal parts, consumer and producer surplus, and the distance gap between two runners read as an area. A Part C adds five classics: sine against sine 2x, the lens between a circle and a parabola, an area equal to ln 2, the Lorenz curve and the retention basin. Sheet
  2. 10 Numerical integration, 201-SN3-RE (201-NYB) What you do when no antiderivative exists: left, right and midpoint rectangle sums, the trapezoidal rule and Simpson's rule with its parity condition, error bounds and the order of convergence (divide by 4 against divide by 16), the number of subintervals needed for a required accuracy, and the two real situations where no other method exists. Also covered: the midpoint rule and the exact relations between the three methods (the midpoint error is half the trapezoidal error with the opposite sign, and Simpson with 2n subintervals comes for free as (2M+T)/3), arc length as the most common use of the method, the machine-precision floor below which refining no longer helps, and the singularities where the error bound becomes empty and only a change of variable saves the computation. A Part C adds five classics: bracketing with left and right rectangles, the sine arch where the trapezoids beat Simpson, a concave function, the cross-section of a river and the pendulum at large amplitudes. Sheet
  3. 11 Applications of integration, 201-SN3-RE (201-NYB) Volumes of revolution by disks and washers, up to a proof of the volume of a sphere; cylindrical shells checked against disks on the same region; arc length and surface area (the sphere again); the average value and the RMS value of alternating current; work, from the spring to the tank being emptied and on to escape velocity; and accumulation from a rate: distance versus displacement, marginal cost, probability. Also covered: volumes by known cross-sections (squares, equilateral triangles, a wedge cut from a cylinder) and Cavalieri's principle, which gives the sphere without an integral; the centre of mass of a plate and Pappus's theorem, which recovers in one line the volumes found by shells (and the torus along the way); the hydrostatic force on a gate, with the dam paradox and the centre of pressure at two thirds of the depth; and four models from other fields: consumer surplus, cardiac output by dye dilution, a rod of variable density and the time to drain a tank by Torricelli's law. A Part C adds five classics: the drilled sphere, washers about a shifted axis, surface areas, the mass of a planet and the work done by a gas. Sheet

Parametric curves and polar coordinates

  1. 12 Parametric curves One calculation exercise first: ten slopes and second derivatives to compute directly on parametric curves, from a polynomial curve to an off-centre ellipse; then the chapter that many course outlines place right after the applications of the integral, and the one most often missing from review: going from a parametrization to a Cartesian equation without losing the domain or the orientation, computing a slope as a quotient of derivatives, recognizing a cusp, finding a double point, then computing the area, the arc length and the area of a surface of revolution directly in terms of the parameter. The cycloid and the astroid are worked through in full. A Part C adds five classics: a parametrized hyperbola, the involute of a circle, the Bézier curve, two drones whose paths cross and the launch of a projectile. Sheet
  2. 13 Polar coordinates, 201-SN3-RE (201-NYB) Polar coordinates and their pitfalls: the arctangent that lands in the wrong quadrant, the circle traced twice when you integrate over a full turn, the rose that has k petals or 2k depending on parity, the bounds of an inner loop that are the zeros of r, and the pole, an intersection that algebra never sees. Area as one half r squared, arc length, the cardioid, the limaçon and the lemniscate worked through in full. A Part C adds five classics: converting to polar form, the spiral of Archimedes, the limaçon and its horizontal tangent, the orbit of a comet and a sprinkler with a variable range. Sheet

Sequences and series

  1. 14 Sequences, 201-SN3-RE (201-NYB) One calculation exercise first: ten limits of sequences to compute directly, from the dominant term to the squeeze theorem; then the gateway to series: explicit or recursive general term, a proper definition of convergence, the techniques for computing limits of sequences, the geometric sequence and its four regimes, the Monotone Convergence Theorem, recursive sequences with the fixed-point trap, and the distinction between a sequence and a series that decides the rest of the semester. Added to these: the three tools for monotonicity (difference, ratio, associated function) with the index where the ratio equals exactly 1, Riemann sums read backwards to find the limit of a sum whose number of terms grows, up to the n-th root of n factorial that tends to one over e, adjacent sequences that pin down e and underpin the bisection method, and attracting or repelling fixed points, where the derivative test finally explains why a recursion converges, oscillates or settles on a cycle. A Part C adds five classics: a daily dose of medication, Fibonacci and the golden ratio, five limits with a variable exponent, paying off a loan and the Koch snowflake. Sheet
  2. 15 Infinite series, 201-SN3-RE (201-NYB) The geometric series proved and applied all the way to the bouncing ball, the divergence test and its false converse, the harmonic series that diverges gently, the integral test linked to improper integrals, the ratio test with its successes and its silent case, alternating series with the error controlled by the first neglected term, and an exam-level classification drill. Added to these: the limit comparison test, which concludes where the inequality falls on the wrong side, up to the series in sine and cosine of one over n; the algebra of infinite sums, with Grandi's series, whose two groupings give 0 and 1, and the exact rule for the sum of two series; the root test, which survives the oscillations that kill the ratio; and the fine boundary beyond p-series, where the series in one over n log n still diverges while the one in one over n log squared already converges. The classics part goes back over the geometric series and its starting index, the divergence test in five verdicts, the choice between comparison and the ratio test, the multiplier effect of public spending, and nested squares. Sheet
  3. 16 Series with sin, cos, exp and ln terms, 201-SN3-RE (201-NYB) A set written entirely at exam level, centred on the case students miss most often: the general term contains a sine, a cosine, an exponential or a logarithm. The ratio test and its two classic limits in k over e, alternating series and conditional convergence, standard asymptotic equivalents, discussion according to a parameter, and estimating the remainder. The classics part adds the standard equivalents in one line each, alternating series built on the usual functions, growth rates placed on the scale, the stack of books that overhangs the drop as far as you like and the alternating harmonic series rearranged until its sum changes. Sheet
  4. 17 Advanced tests: root, condensation, Raabe, Abel, 201-SN3-RE (201-NYB) The third set on infinite series, the one that closes the gaps: the Cauchy criterion as a necessary and sufficient condition, the condensation test that proves the p-series and Bertrand results in one line, how the root test compares in strength with the ratio test, Raabe's test when the ratio tends to 1, Abel's test for series that are not alternating, the Cauchy product, and above all the major pitfall: an asymptotic equivalent can only be used on terms of constant sign. The part on the classics adds telescoping sums with square roots, the inverse tangent and factorials, Raabe's test on products, Abel's test in practice, the Cauchy product of two geometric distributions, and how to speed up the alternating harmonic series by averaging two partial sums. Sheet
  5. 18 Power series and Taylor series, 201-SN3-RE (201-NYB) The radius of convergence by the Ratio Test, with the endpoints tested one at a time, the geometric series promoted to a power series and put to work (term-by-term differentiation and integration all the way to ln 2 and pi/4), building the Maclaurin series of the exponential, sine and cosine, Taylor polynomials hugging the curve, approximation with a controlled error, impossible integrals computed with series, and limits that fall in one line. Also covered: building series by multiplication, composition and division without computing a single derivative (up to the tangent and the binomial series); the Lagrange remainder, which bounds the error when the series is not alternating and fixes the number of terms needed for six decimals of e; the radius of convergence explained by complex singularities, with a function that is smooth on the whole real line and yet whose series stops at 1, and an infinitely differentiable function that its own Taylor series does not represent; and exact sums computed by evaluating power series. A Part C adds five classics: standard radii and intervals of convergence, expansions built by substitution with high-order derivatives read off a coefficient, approximations with a guaranteed error, relativistic kinetic energy at low speed and a fast series that computes ln 3 and ln 2. Sheet

Differential equations

  1. 19 Differential equations Verifying a solution and solving separable equations all the way to the family of circles, the integrating factor for linear equations with the bridge to the RC circuit, exponential growth and decay (carbon 14, bacteria, compound interest), Newton's law of cooling on the exact curve, the brine tank that settles into its steady state, and the logistic model with its inflection point at half capacity. Also covered: the qualitative analysis that predicts everything without solving (slope field, phase line, stability of equilibria, Euler's method and its concavity error), the substitutions that linearize homogeneous equations and the Bernoulli equation, the latter finally proving the logistic solution taken for granted earlier, the orthogonal trajectories of circles and parabolas, falling with air resistance and its terminal velocity that forgets the initial state, and fishing with its collapse threshold, where the maximum sustainable yield coincides exactly with the maximum growth at the inflection point. The classics part adds five separable equations up to the loss of uniqueness, the integrating factor on five linear equations, Euler's and Heun's methods against the exact solution, draining a tank by Torricelli's law and a chemical reaction solved with partial fractions. To open, ten equations to solve in quick succession: five first-order equations whose type must first be recognized (separable, linear, homogeneous or linear in x) and five second-order equations, integrated twice or reduced to first order. Sheet

When the chapters are done

A full paper under exam conditions, on the whole course. Do it once every chapter is behind you: it measures what you can do on a question you have never seen.

Practice exam

Practice final exam: all of integral calculus, 201-SN3-RE (201-NYB)

A three-hour final exam that runs through the whole integral calculus course: integration techniques, indeterminate forms, improper integrals, volumes and applications, sequences, numerical series and power series, Taylor polynomials and differential equations. No question tells you which technique to use.

  • 12 questions
  • 100 points
  • 180 minutes
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