Unit 5 of 10

Integrals

  • 28 lessons
  • Parts 89 to 144

Integration runs in two directions that turn out to be the same idea. Antiderivatives undo differentiation, and definite integrals add up infinitely many tiny pieces to give areas, totals and accumulated change. The fundamental theorem of calculus connects the two.

This is the largest unit. It moves from antiderivatives and Riemann sums to the fundamental theorem, through the main integration techniques and improper integrals, and ends with applications such as areas, volumes, arc length, work and probability.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Indefinite Integrals

  1. Introduction to Indefinite Integrals

    What an antiderivative is, why all antiderivatives on an interval differ by a constant, how a starting value picks out one of them, and how to check an answer by differentiating.

    Theory part 89Practice part 90

  2. Integration Rules for Indefinite Integrals

    The basic rules for finding indefinite integrals, from constants and powers to logarithms, exponentials and trigonometric functions, together with the intervals on which each one holds.

    Theory part 91Practice part 92

Areas as Limits of Sums

  1. Sums and Sigma Notation

    How to write long sums compactly with sigma notation, shift their index, split them with the sum rules, and evaluate sums of powers and geometric sums in closed form.

    Theory part 93Practice part 94

  2. The Area Problem

    How to give the area under a curve an exact meaning by trapping it between sums of rectangles and letting the rectangles get thinner.

    Theory part 95Practice part 96

Definite Integrals

  1. Partitions and Riemann Sums

    How to cut an interval into pieces of any widths, build a Riemann sum from any choice of sample points, and trap every such sum between lower and upper Darboux sums.

    Theory part 97Practice part 98

  2. Introduction to Definite Integrals

    What the definite integral is as the limit of Riemann sums when the mesh goes to zero, which functions have one, how to read its notation, and why it measures signed area.

    Theory part 99Practice part 100

  3. Properties of Definite Integrals

    The rules that let you split, combine, compare and bound definite integrals, use symmetry, handle functions with jumps, and find the average value of a continuous function.

    Theory part 101Practice part 102

The Fundamental Theorem of Calculus

  1. Accumulation Functions and the Fundamental Theorem, Part 1

    Why the integral of a continuous function from a fixed point up to x is a function of x whose derivative is the integrand, and how to differentiate integrals with variable bounds.

    Theory part 103Practice part 104

  2. The Fundamental Theorem, Part 2, and Net Change

    How to evaluate a definite integral exactly with any antiderivative, which hypotheses make that legal, and how integrating a rate gives net change, displacement and total distance.

    Theory part 105Practice part 106

Integration Techniques

  1. The Substitution Method

    How to undo the chain rule by changing variables, for indefinite integrals with back-substitution and for definite integrals with new bounds, and which conditions make the change valid.

    Theory part 107Practice part 108

  2. Integration by Parts

    How the product rule turns into a formula for integrating products, how boundary terms work in definite integrals, and how repeating the method leads to reduction formulas.

    Theory part 109Practice part 110

  3. Integration of Rational Functions

    How to integrate any quotient of polynomials by dividing first when needed, splitting into partial fractions, and integrating each simple piece on intervals where the denominator is not zero.

    Theory part 111Practice part 112

  4. Trigonometric Integrals

    How to integrate products of powers of sine and cosine, or of tangent and secant, by choosing a substitution from the parity of the exponents, and how product-to-sum identities handle different frequencies.

    Theory part 113Practice part 114

  5. Trigonometric and Inverse Substitutions

    How to remove square roots of sums and differences of squares by setting x equal to a sine, tangent or secant, choosing the branch so the root simplifies correctly, and returning to x with a triangle.

    Theory part 115Practice part 116

  6. Hyperbolic and Tangent-Half-Angle Substitutions

    How the substitutions x = a sinh t, x = a cosh t and t = tan(x/2) clear square roots and trigonometric fractions from integrals, and on which intervals each one is valid.

    Theory part 117Practice part 118

  7. Other Integration Techniques and Symbolic Tools

    How to guess an antiderivative's form and solve for its coefficients, when a differential binomial has an elementary antiderivative, and how to use tables and computer algebra safely.

    Theory part 119Practice part 120

Improper Integrals

  1. Improper Integrals over Unbounded Intervals

    How an integral over an infinite interval is defined as a limit, when it converges, which power functions have finite area under their tails, and why a symmetric limit is not enough.

    Theory part 121Practice part 122

  2. Improper Integrals with Singularities and Comparison Tests

    How to integrate a function that blows up inside or at the edge of an interval, and how to decide convergence by comparing with integrals you already know, without computing anything exactly.

    Theory part 123Practice part 124

Applications of Integration

  1. Area Between Two Curves

    How to find the area of a region bounded by curves, by locating where they meet, choosing vertical or horizontal slices, and splitting the integral wherever the top and bottom curves swap.

    Theory part 125Practice part 126

  2. Average Value of a Function

    How to average a function over an interval with an integral, why that is usually not the value at the midpoint, where a continuous function attains its average, and how weights change the average.

    Theory part 127Practice part 128

  3. Volumes by Cross-Sections, Disks, and Washers

    How to find a volume by integrating the area of its slices, which explains the formulas for cylinders, cones, pyramids and balls and gives the disk and washer methods for solids of revolution.

    Theory part 129Practice part 130

  4. Volumes by Cylindrical Shells

    How to find the volume of a solid of revolution by adding thin cylindrical shells, how to read the radius and height of a shell, and when shells are easier than washers.

    Theory part 131Practice part 132

  5. Arc Length of Function Graphs

    How the length of a curved graph is defined through inscribed polygons, why it equals the integral of the square root of one plus the squared derivative, and what to do when that integral has no formula.

    Theory part 133Practice part 134

  6. Surface Area of Revolution

    How the area of a surface made by spinning a curve around an axis is built from thin cone bands, giving an integral of circumference times arc length, and which conditions make that integral valid.

    Theory part 135Practice part 136

  7. Mass, Moments, and Centres of Mass

    How to find the mass and balance point of a rod with varying density and the centroid of a flat region, and how the centroid gives volumes and surface areas of revolution in one step.

    Theory part 137Practice part 138

  8. Work, Energy, and Hydrostatic Force

    How to compute the work done by a force that changes along the way, the work needed to stretch springs and pump liquids, and the force water puts on a submerged wall.

    Theory part 139Practice part 140

  9. Integration in Economics, Ecology, and Physiology

    How integrals measure consumer and producer surplus, the present value of income, changes in a renewable stock, blood flow and cardiac output, and what each model assumes.

    Theory part 141Practice part 142

  10. Integration in Probability Theory

    How a probability density turns probabilities into areas, and how integrals give the distribution function, the mean, the variance and the standard deviation, when they exist.

    Theory part 143Practice part 144