Calculus

Integrals From Easy to Advanced

One long visual course in 30 chapters that runs from area and Riemann sums through the main integration techniques to multiple integrals, flux and contour integration.

  • Intermediate
  • 30 chapters
  • Free
Browse the lessons

The videos are coming soon

Every lesson already has its own page with precise definitions, a worked example for each key term and short exercises with answers. The videos will appear on the lesson pages as they come out.

About this course

An integral adds up small contributions, an area from thin strips, a volume from slices or the flux of a field from small patches of a surface. Each chapter of this course takes one integral and the method that evaluates it, starting from the area of a rectangle and building through Riemann sums, the fundamental theorem and the symbolic techniques to symmetry, improper integrals, numerical approximation and power series, then areas, volumes, arc length and surface area, double and triple integrals in new coordinates, line integrals, Green’s theorem and the divergence theorem, and finally the Gaussian integral, differentiation under the integral sign and contour integration with residues. Every chapter shows how to recognize that the method fits, works the integral step by step with the conditions it needs and checks the answer.

The course is one long video in 30 chapters rather than a set of separate lessons. Each chapter introduces one new main idea and draws on earlier ones, and the outline shows what each chapter builds on, so you can follow the whole progression or return to a single method. The video is coming soon. Every chapter has its own page with precise definitions, a worked example for each key term and a few exercises with answers, so you can start reading now.

The reference sheet collects the integrals, methods and conditions chapter by chapter, with links to the pages that explain them. Basic algebra, graphs and some familiarity with derivatives are enough to start, and new ideas such as double integrals, vector fields and complex numbers are introduced when a chapter first needs them.

Other courses on the site are optional background or further study, not prerequisites. Foundations for Calculus reviews the algebra the techniques rely on, such as completing the square, partial fractions as an algebraic identity and rewriting trigonometric powers. Calculus develops derivatives, definite integrals, the integration techniques, improper integrals, the applications of integration and power series in full. Multivariable and Vector Calculus treats double and triple integrals, the Jacobian, line integrals, Green’s theorem, Stokes’ theorem and the divergence theorem. Differential Equations puts integration to work in separable and exact equations, integrating factors and the Laplace transform, which is defined by an improper integral.

Good to know first

  • Basic algebra, including factoring and expanding squares
  • Graphs and coordinates
  • Some familiarity with derivatives
  • The chain rule and the product rule
  • Basic trigonometry and its identities

Course outline

The video for all 30 chapters is coming soon. The course will be one long video, with a chapter for each of them. Every chapter already has its own page, so open a unit to see its chapters, or follow a chapter for its definitions and examples.

  1. Accumulation and foundations

    4 chapters coming soon
    Open the Accumulation and foundations unit page

    Area, Riemann sums, the fundamental theorem and linearity

    1. What an integral measures, starting with a rectangleComing soon

      Chapter 1

      definite integral, integrand, limits of integration, signed area, accumulation and units of an integral

      Also uses multiplication and coordinates on a graph.

    2. The Riemann-sum limit for a changing heightComing soon

      Chapter 2

      Riemann sum, regular partition, mesh width, right-endpoint sum, left-endpoint sum and limit of Riemann sums

      Builds on What an integral measures, starting with a rectangle. Also uses arithmetic series sums and limits as n increases.

    3. The fundamental theorem and the power rule for a parabolaComing soon

      Chapter 3

      fundamental theorem of calculus, accumulation function, antiderivative, reverse power rule, indefinite integral and constant of integration

      Builds on The Riemann-sum limit for a changing height. Also uses the derivative of x cubed.

    4. Linearity and signed contributions for a sum of termsComing soon

      Chapter 4

      linearity of the integral, sum rule for integrals, constant multiple rule, signed contribution and average value of a function

      Builds on What an integral measures, starting with a rectangle, The Riemann-sum limit for a changing height and The fundamental theorem and the power rule for a parabola.

  2. Recognizing symbolic methods

    6 chapters coming soon
    Open the Recognizing symbolic methods unit page

    Substitution, integration by parts and trigonometric identities

    1. Substitution for a hidden inner variableComing soon

      Chapter 5

      u-substitution, inner function, chain rule in reverse, differential du, changing the bounds and one-to-one substitution

      Builds on The fundamental theorem and the power rule for a parabola. Also uses the chain rule and the derivative of sine.

    2. Integration by parts for a useful productComing soon

      Chapter 6

      integration by parts, product rule in reverse, boundary term, choosing u and dv and complementary area

      Builds on The fundamental theorem and the power rule for a parabola. Also uses the product rule and the derivative of e to the x.

    3. Use a half-angle identity for a squared sineComing soon

      Chapter 7

      half-angle identity, power-reducing identity, integral of sine squared, cosine of a doubled angle and cancellation over a full period

      Builds on The fundamental theorem and the power rule for a parabola and Linearity and signed contributions for a sum of terms. Also uses the Pythagorean identity and the double-angle formula for cosine.

    Trigonometric substitution, partial fractions and completing the square

    1. Trigonometric substitution for a circle inside a square rootComing soon

      Chapter 8

      trigonometric substitution, sine substitution, circle parameterization, angle bounds, nonnegative square root and inverse sine

      Builds on Substitution for a hidden inner variable and Use a half-angle identity for a squared sine. Also uses circle coordinates and right-triangle trigonometry.

    2. Partial fractions for a factorable denominatorComing soon

      Chapter 9

      partial fraction decomposition, rational function, comparing coefficients, logarithmic antiderivative and poles outside the interval

      Builds on Linearity and signed contributions for a sum of terms. Also uses factoring a difference of squares and the derivative of the logarithm of an absolute value.

    3. Completing the square for a shifted quadraticComing soon

      Chapter 10

      completing the square, irreducible quadratic, negative discriminant, inverse tangent antiderivative and shift and scale substitution

      Builds on Substitution for a hidden inner variable. Also uses expanding a square and the derivative of inverse tangent.

  3. Structure, limits and approximation

    4 chapters coming soon
    Open the Structure, limits and approximation unit page

    Symmetry, improper integrals, Simpson's rule and power series

    1. Read the symmetry before calculatingComing soon

      Chapter 11

      symmetry rule for definite integrals, odd function, even function, symmetric interval and parity

    2. When an infinite region has finite areaComing soon

      Chapter 12

      improper integral, convergent integral, divergent integral, p-test for integrals, boundary case p equals 1 and remaining tail

    3. Approximate an integral with a controlled errorComing soon

      Chapter 13

      composite Simpson's rule, numerical integration, quadratic interpolant, Simpson's error bound, fourth derivative bound and no elementary antiderivative

    4. Integrate an infinite expansionComing soon

      Chapter 14

      power series, exponential series, term-by-term integration, uniform convergence, removable singularity and geometric tail bound

  4. Geometric construction

    5 chapters coming soon
    Open the Geometric construction unit page

    Area between curves and volumes by washers and shells

    1. Build the integral for a region between curvesComing soon

      Chapter 15

      area between two curves, intersection points, top minus bottom, vertical and horizontal strips and absolute difference

    2. Build a solid from washersComing soon

      Chapter 16

      washer method, solid of revolution, volume by slicing, cross-sectional area and outer and inner radius

    3. Build a solid from cylindrical shellsComing soon

      Chapter 17

      shell method, cylindrical shell, shell radius and height, rotation about the y-axis and strip parallel to the axis

    Arc length and surface area of revolution

    1. Measure the length of a curveComing soon

      Chapter 18

      arc length, arc length integral, arc length element, Pythagorean length of a short segment and continuously differentiable curve

    2. Measure a curved surface in three dimensionsComing soon

      Chapter 19

      surface area of revolution, surface area element, conical frustum, slant length and lateral surface

  5. Multiple integration and coordinates

    5 chapters coming soon
    Open the Multiple integration and coordinates unit page

    Double integrals and the order of integration

    1. Accumulate over two coordinatesComing soon

      Chapter 20

      double integral, iterated integral, volume under a surface, rectangular region, inner and outer integrals and holding one variable fixed

    2. Change the order, reveal the missing factorComing soon

      Chapter 21

      reversing the order of integration, Fubini's theorem, triangular region, bounds read from a sketch and missing factor for substitution

      Builds on Substitution for a hidden inner variable and Accumulate over two coordinates.

    Polar coordinates, the Jacobian and spherical coordinates

    1. Let the circle choose the coordinatesComing soon

      Chapter 22

      polar coordinates, double integral in polar coordinates, polar area element, annular sector and radial symmetry

      Builds on Accumulate over two coordinates and Change the order, reveal the missing factor.

    2. Build coordinates from the boundariesComing soon

      Chapter 23

      linear change of variables, Jacobian determinant, area scale factor, inverse map, absolute value of the Jacobian and parallelogram region

      Builds on Accumulate over two coordinates and Let the circle choose the coordinates.

    3. A ball, a radial density, and a volume cellComing soon

      Chapter 24

      triple integral, spherical coordinates, spherical volume element, polar angle and azimuth, radial density and mass from density

      Builds on Let the circle choose the coordinates and Build coordinates from the boundaries.

  6. Fields, circulation and flux

    3 chapters coming soon
    Open the Fields, circulation and flux unit page

    Line integrals, Green's theorem and the divergence theorem

    1. When a path integral only needs its endpointsComing soon

      Chapter 25

      line integral, fundamental theorem for line integrals, path independence, conservative vector field, potential function and exact differential

      Builds on The fundamental theorem and the power rule for a parabola and Accumulate over two coordinates.

    2. Add the turning inside, find the circulation outsideComing soon

      Chapter 26

      Green's theorem, circulation, scalar curl, counterclockwise orientation and cancellation along shared edges

      Builds on When a path integral only needs its endpoints.

    3. Count outward flow from the sources insideComing soon

      Chapter 27

      divergence theorem, flux, surface integral, divergence, outward unit normal and cancellation across shared faces

      Builds on A ball, a radial density, and a volume cell and Add the turning inside, find the circulation outside.

  7. Advanced transformations

    3 chapters coming soon
    Open the Advanced transformations unit page

    The Gaussian integral, parameter differentiation and contour integration

    1. Square the question, reveal a circleComing soon

      Chapter 28

      Gaussian integral, squaring the integral, product of independent integrals, improper double integral and polar coordinates on the whole plane

      Builds on When an infinite region has finite area, Approximate an integral with a controlled error and Let the circle choose the coordinates.

    2. Make a family, then differentiate the familyComing soon

      Chapter 29

      differentiation under the integral sign, Leibniz integral rule, parameter family of integrals, integrable bound, removable endpoint limits and constant from a known value

      Builds on The fundamental theorem and the power rule for a parabola, Substitution for a hidden inner variable and When an infinite region has finite area.

    3. Leave the real line, collect the poles, returnComing soon

      Chapter 30

      contour integration, residue theorem, Cauchy's theorem, analytic function, simple pole and its residue, upper half-plane semicircle and vanishing arc bound

      Builds on When an infinite region has finite area and Add the turning inside, find the circulation outside.