APEX Calculus - Multiple Integration

This course develops multiple integration through double and triple integrals, including iterated integration, changing integration order, polar, cylindrical, and spherical coordinates, Jacobians, and change of variables. Worked examples calculate volumes of tetrahedra, cones, spherical wedges, and conical cups.

Syllabus

Introducing the double integral.
Defining the double integral.
Iterated integrals and Fubini's theorem.
Examples of integrals over rectangles.
Integrals over general regions.
Example: integral over a general region.
Reversing the order of integration: Example 1.
Reversing the order of integration: Example 2.
Double integrals in polar coordinates.
Polar coordinates: Examples 1 and 2.
Polar coordinates: Example 3.
Example: volume of a tetrahedron.
Example: volume using polar coordinates.
Introduction to triple integrals.
Tetrahedron volume, as a triple integral.
Changing order of integration in a triple integral.
Volume problem leading to cylindrical coordinates.
The cylindrical coordinate system.
Cylindrical coordinates and volume of a cone.
Introducing spherical coordinates.
Integration in spherical coordinates.
Volume of a spherical wedge.
Volume of a conical cup.
The spherical coordinate Jacobian.
Transformations and change of variables.
Change of variables example: parallelogram.
Working with the inverse transformation (a shortcut!).
Another change of variables example.