First-year courses · Mathematics

Calculus II: integration techniques and uses

Riemann sums, the Fundamental Theorem, substitution, parts, partial fractions, trig substitution, improper integrals, areas and volumes in flashcards.

44 cards8 topicsAbout 20 minutes a roundTypical first-year Calculus II (integral calculus), integration and its applications

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About this deck

This deck covers integral calculus as it is usually taught at the end of Calculus I and through most of Calculus II in Canadian first-year programs. It starts with antiderivatives and Riemann sums, moves through both parts of the Fundamental Theorem of Calculus, then works through the main techniques: substitution, integration by parts, trig integrals, trig substitution and partial fractions. It finishes with improper integrals and applications: area between curves, volumes by disks, washers and shells, arc length and average value.

You need fluent differentiation (every integral formula is a derivative formula read backwards), basic trig identities such as sin^2 x + cos^2 x = 1 and the double-angle formulas, and comfort with logarithm and exponent laws.

Integration is a recognition skill. When a technique card comes up, try to name the technique and the first line of working before you flip it, then check the example by differentiating your answer. Spend extra time on the mixed review at the end of each week: shuffle the Substitution, Integration by parts and Partial fractions tags together so you learn to choose a method rather than just apply one you were told to use.

Typical mistakes: forgetting + C on indefinite integrals; changing variables in a definite integral but keeping the old limits; dropping the absolute value in ln|x|; and treating an improper integral as ordinary, for example integrating 1/x^2 from -1 to 1 straight through the discontinuity at 0 and getting a negative number for a positive function.

Before you start: Calculus I (limits and derivatives) and Grade 12 trigonometry, including identities.

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Antiderivatives and Riemann sums

Antiderivative: A function F whose derivative is f on an interval. Any two of them for the same f differ by a constant.Example: x^3, x^3 + 5 and x^3 - 2 all have derivative 3x^2.
Indefinite integral: The family of all antiderivatives of f, written integral of f(x) dx = F(x) + C, where C is an arbitrary constant.Example: integral of cos x dx = sin x + C.
Power rule for integrals: integral of x^n dx = x^(n+1)/(n + 1) + C for any constant n other than -1.Example: integral of x^4 dx = x^5/5 + C; integral of x^(-1/2) dx = 2 sqrt(x) + C.
Integral of 1/x: integral of (1/x) dx = ln|x| + C; this fills the gap in the power rule at n = -1.Example: integral of 3/x dx = 3 ln|x| + C.
Basic exponential and trig integrals: integral of e^x dx = e^x + C; of sin x dx = -cos x + C; of cos x dx = sin x + C; of sec^2 x dx = tan x + C.Example: integral of (e^x + sin x) dx = e^x - cos x + C.
Riemann sum: An approximation of area: sum of f(x_i*) times delta x over n subintervals of width delta x = (b - a)/n, using a sample point x_i* in each.Example: x^2 on [0, 2], n = 4, right endpoints: 0.5(0.25 + 1 + 2.25 + 4) = 3.75 (exact value 8/3).
Left, right and midpoint sums: Sample points at left ends, right ends or centres of subintervals. For an increasing function, left sums underestimate and right sums overestimate.Example: x^2 on [0, 2], n = 4, midpoints: 0.5(0.0625 + 0.5625 + 1.5625 + 3.0625) = 2.625.
Sigma sum formulas: Sum of i for i = 1 to n is n(n + 1)/2; sum of i^2 is n(n + 1)(2n + 1)/6; sum of i^3 is (n(n + 1)/2)^2.Example: Sum of 1 to 10 is 55; sum of squares 1 to 10 is 385.

Definite integrals and the FTC

Definite integral: The limit of Riemann sums as the number of subintervals goes to infinity; it gives the signed area between the graph and the x-axis from a to b.Example: integral of x^2 from 0 to 2 = 8/3.
Properties of definite integrals: Swapping the limits changes the sign; integrals over adjacent intervals add (a to b plus b to c equals a to c); the integral from a to a is 0.Example: If integral from 0 to 3 of f is 5 and from 0 to 1 is 2, then from 1 to 3 it is 3.
Net signed area versus total area: Area below the x-axis counts negative in a definite integral; total area integrates |f| instead.Example: integral of sin x from 0 to 2 pi is 0, but the total area enclosed is 4.
Fundamental Theorem of Calculus, part 1: If f is continuous, then g(x) = integral of f(t) dt from a to x is differentiable and g'(x) = f(x): differentiating an accumulation function returns the integrand.Example: d/dx of integral of cos(t^2) dt from 0 to x equals cos(x^2).
FTC part 1 with a variable upper limit: When the upper limit is a function u(x), the chain rule applies: d/dx of the integral from a to u(x) of f(t) dt = f(u(x)) u'(x).Example: d/dx of integral of ln t dt from 1 to x^3 = ln(x^3) * 3x^2 = 9x^2 ln x.
Fundamental Theorem of Calculus, part 2: If f is continuous on [a, b] and F is any antiderivative of f, then the integral of f from a to b equals F(b) - F(a).Example: integral of 2x dx from 1 to 3 = 3^2 - 1^2 = 8.
Net change theorem: Integrating a rate of change over an interval gives the total net change of the quantity over that interval.Example: Velocity v(t) = 3t^2 m/s from t = 0 to 2 s gives displacement 2^3 - 0 = 8 m.
Displacement versus distance travelled: Integrating v(t) gives displacement (net change in position); integrating |v(t)| gives total distance travelled.Example: v = t - 1 on [0, 2]: displacement 0 m, distance 1 m.
Even and odd symmetry: For odd f, the integral from -a to a is 0; for even f, it equals twice the integral from 0 to a.Example: integral of x^3 from -2 to 2 is 0; integral of x^2 from -2 to 2 is 2(8/3) = 16/3.

Substitution

u-substitution: Reverses the chain rule: if u = g(x), then integral of f(g(x)) g'(x) dx = integral of f(u) du.Example: integral of 2x cos(x^2) dx: u = x^2 gives integral of cos u du = sin(x^2) + C.
Substitution in a definite integral: After setting u = g(x), change the limits to g(a) and g(b) so you never need to go back to x.Example: integral of 2x(x^2 + 1)^3 from 0 to 1: u runs 1 to 2, integral of u^3 du = (16 - 1)/4 = 15/4.
Integrals of the form f'(x)/f(x): When the numerator is the derivative of the denominator, the result is ln|f(x)| + C (substitute u = f(x)).Example: integral of tan x dx = integral of sin x/cos x dx = -ln|cos x| + C.

Integration by parts

Integration by parts: From the product rule: integral of u dv = uv - integral of v du. Choose u so that du is simpler and dv so that it can be integrated.Example: integral of x e^x dx: u = x, dv = e^x dx, giving x e^x - e^x + C.
LIATE guideline: A rule of thumb for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential; pick the type that appears first.Example: integral of x ln x dx: u = ln x gives (x^2/2) ln x - x^2/4 + C.
Repeated integration by parts: Some integrals need parts more than once, each time lowering the power of a polynomial factor.Example: integral of x^2 e^x dx = e^x (x^2 - 2x + 2) + C.
Cyclic integration by parts: When applying parts twice brings back the original integral, move it to the other side of the equation and solve for it.Example: integral of e^x sin x dx = e^x (sin x - cos x)/2 + C.
Integral of ln x: Use parts with u = ln x and dv = dx: the result is x ln x - x + C.Example: integral of ln x from 1 to e = (e - e) - (0 - 1) = 1.

Trig integrals and trig substitution

Powers of sine and cosine: With an odd power, save one factor and convert the rest using sin^2 + cos^2 = 1; with only even powers, use half-angle identities.Example: integral of sin^2 x dx = integral of (1 - cos 2x)/2 dx = x/2 - sin(2x)/4 + C.
Substitution x = a sin(theta): Used for sqrt(a^2 - x^2); it turns the root into a cos(theta).Example: integral of dx/sqrt(4 - x^2) = arcsin(x/2) + C.
Substitution x = a tan(theta): Used for sqrt(a^2 + x^2) or a^2 + x^2; it turns the root into a sec(theta).Example: integral of dx/(x^2 + 9) = (1/3) arctan(x/3) + C.
Substitution x = a sec(theta): Used for sqrt(x^2 - a^2) with x >= a; it turns the root into a tan(theta).Example: integral of dx/(x sqrt(x^2 - 1)) = arcsec(x) + C for x > 1.

Partial fractions

Partial fraction decomposition: Rewriting a proper rational function as a sum of simpler fractions, one or more for each factor of the denominator, so each piece can be integrated.Example: 1/(x^2 - 1) = (1/2)/(x - 1) - (1/2)/(x + 1), so the integral is (1/2) ln|(x - 1)/(x + 1)| + C.
Repeated linear factor: A factor (x - a)^k needs one term for each power: A1/(x - a) + A2/(x - a)^2 + ... + Ak/(x - a)^k.Example: (x + 3)/(x - 1)^2 = 1/(x - 1) + 4/(x - 1)^2.
Irreducible quadratic factor: A quadratic factor that cannot be factored over the reals gets a numerator of the form Ax + B; it usually integrates to a log plus an arctan.Example: 1/(x(x^2 + 1)) = 1/x - x/(x^2 + 1).
Long division before partial fractions: If the degree of the numerator is at least the degree of the denominator, divide first to get a polynomial plus a proper fraction.Example: x^2/(x + 1) = x - 1 + 1/(x + 1).

Improper integrals

Improper integral over an infinite interval: Defined as a limit: integral from a to infinity of f = limit as t approaches infinity of the integral from a to t. It converges if that limit is finite.Example: integral of 1/x^2 from 1 to infinity = lim (1 - 1/t) = 1.
Improper integral with an unbounded integrand: If f blows up at an endpoint (or inside the interval), replace that point with a variable and take a one-sided limit.Example: integral of 1/sqrt(x) from 0 to 1 = lim as t approaches 0+ of (2 - 2 sqrt(t)) = 2.
p-test for integrals: integral of 1/x^p from 1 to infinity converges when p > 1 and diverges when p <= 1.Example: p = 2 converges (value 1); p = 1 diverges because ln t grows without bound.
Comparison test for integrals: If 0 <= f <= g on [a, infinity) and the integral of g converges, so does that of f; if the integral of f diverges, so does that of g.Example: For x >= 1, e^(-x^2) <= e^(-x), whose integral converges, so the integral of e^(-x^2) from 1 to infinity converges.

Applications

Area between curves: integral from a to b of (top function - bottom function) dx, where a and b are usually the intersection points.Example: Between y = x and y = x^2 on [0, 1]: integral of (x - x^2) = 1/2 - 1/3 = 1/6.
Disk method: Volume from rotating a region about an axis with no gap: V = pi times integral of R(x)^2 dx, where R is the radius of each circular slice.Example: y = sqrt(x) on [0, 4] about the x-axis: pi integral of x dx = 8 pi.
Washer method: Volume of a solid with a hole: V = pi times integral of (R^2 - r^2) dx, using outer radius R and inner radius r.Example: Region between y = x and y = x^2 about the x-axis: pi integral of (x^2 - x^4) from 0 to 1 = 2 pi/15.
Shell method: Volume from rotating about a vertical axis using cylindrical shells: V = 2 pi times integral of (radius)(height) dx.Example: Region under y = x^2 on [0, 1] about the y-axis: 2 pi integral of x^3 dx = pi/2.
Average value of a function: The mean height of f on [a, b]: (1/(b - a)) times the integral of f from a to b.Example: x^2 on [0, 3]: (1/3)(9) = 3.
Mean Value Theorem for integrals: If f is continuous on [a, b], it actually reaches its average value: f(c) equals the average for some c in [a, b].Example: x^2 on [0, 3] has average 3, reached at c = sqrt(3).
Arc length: Length of y = f(x) from a to b: integral of sqrt(1 + (f'(x))^2) dx.Example: y = (2/3)x^(3/2) on [0, 3]: f' = sqrt(x), length = integral of sqrt(1 + x) = (2/3)(8 - 1) = 14/3.