Unit 4 of 6

Multiple Integrals

  • 9 lessons
  • Parts 47 to 64

A double integral adds up a quantity over a region of the plane, and a triple integral over a solid. This unit defines them through Riemann sums, evaluates them as iterated integrals, and sets up the bounds for general regions.

Polar, cylindrical and spherical coordinates simplify regions with circular symmetry, and the Jacobian explains the general change of variables. Applications include area, mass, centers of mass and probability, and the unit closes with improper integrals and reversing the order of integration.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Double Integrals

  1. Double Integrals over Rectangles

    How a double integral over a rectangle is built from area weighted sums, what it means as signed volume and average value, and when it can be computed as an iterated integral in either order.

    Theory part 47Practice part 48

  2. Double Integrals over General Regions

    How to turn a region bounded by curves into correct iterated bounds, how to reverse the order of integration by redrawing the region, and when a double integral is improper.

    Theory part 49Practice part 50

  3. Double Integrals in Polar Coordinates

    Where the extra factor r in polar double integrals comes from, and how to choose radius and angle bounds that cover a disk, ring, sector or shifted circle exactly once.

    Theory part 51Practice part 52

  4. Applications of Double Integrals

    How double integrals give the area, mass, balance point and turning resistance of a thin plate, and the probabilities, averages and covariance of a pair of random quantities.

    Theory part 53Practice part 54

Triple Integrals

  1. Triple Integrals in Rectangular Coordinates

    How a triple integral adds up density times volume over a solid, how to set up its bounds from a projection and vertical segments, and how it gives volume, mass, centers of mass and averages.

    Theory part 55Practice part 56

  2. Triple Integrals in Cylindrical Coordinates

    Why the volume element in cylindrical coordinates is r dr dθ dz, and how to read bounds for solids built from cylinders, cones and paraboloids, including changing the order and using symmetry.

    Theory part 57Practice part 58

  3. Triple Integrals in Spherical Coordinates

    How spherical coordinates with the polar angle measured from the positive z-axis describe balls, cones, caps and shifted spheres, and why the volume element is ρ² sin φ dρ dφ dθ.

    Theory part 59Practice part 60

Changing Variables and Improper Integrals

  1. Change of Variables and the Jacobian

    Why the absolute value of the Jacobian determinant measures how a substitution stretches area and volume, and which local and global conditions make a change of variables in a multiple integral valid.

    Theory part 61Practice part 62

  2. Improper Multiple Integrals and Order of Integration

    How to define a double integral over an unbounded region or near a singularity as a limit over growing regions, when the order of integration may be swapped, and how this proves the Gaussian integral.

    Theory part 63Practice part 64