Unit 5 of 7

Multiple integration and coordinates

  • 5 chapters
  • Chapters 20 to 24

Double and triple integrals are built from columns and slices. Reversing the order of integration can help, and polar, linear and spherical coordinates each bring their own area or volume scale.

A double integral over a square adds columns, and an iterated integral adds slices one direction at a time. Over a triangle, reversing the order of integration turns an inner integral with no elementary antiderivative into one that a substitution finishes, with Fubini's theorem behind the change. A disk calls for polar coordinates, whose area element carries an extra factor r. A parallelogram whose sides keep x plus y or x minus y constant uses those two expressions as coordinates, with the Jacobian determinant as the area scale. A ball with a radial density calls for spherical coordinates.

Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.

Double integrals and the order of integration

  1. Accumulate over two coordinates

    How a double integral over a rectangular region adds thin columns to give the volume under a surface, and how an iterated integral computes it through inner and outer integrals by holding one variable fixed.

    Chapter 20

  2. Change the order, reveal the missing factor

    How reversing the order of integration over a triangular region, with new bounds read from a sketch and justified by Fubini's theorem, supplies the missing factor that a substitution needs.

    Chapter 21

Polar coordinates, the Jacobian and spherical coordinates

  1. Let the circle choose the coordinates

    How polar coordinates turn a double integral over a disk into an iterated integral in radius and angle, why the polar area element carries an extra factor r from a thin annular sector, and how radial symmetry helps.

    Chapter 22

  2. Build coordinates from the boundaries

    How a linear change of variables turns a parallelogram region into a square, why the area element carries the absolute value of the Jacobian determinant of the inverse map, and how that area scale factor is found.

    Chapter 23

  3. A ball, a radial density, and a volume cell

    How a triple integral gives the mass of a ball with a radial density, how spherical coordinates use a polar angle and an azimuth, and why the spherical volume element is r² sin φ dr dφ dθ.

    Chapter 24