The course covers learning outcomes and goals related to multiple integration, including introducing and defining double integrals, understanding iterated integrals, applying Fubini's theorem, working with integrals over rectangles and general regions, utilizing polar coordinates for double integrals, introducing and applying triple integrals, exploring cylindrical and spherical coordinates, understanding transformations and change of variables. The course teaches individual skills such as calculating volumes of various shapes using different coordinate systems. The teaching method includes video lectures with examples and explanations. The intended audience for this course is students studying advanced calculus, specifically those interested in mastering multiple integration techniques.
Overview
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.
Taught by
Sean Fitzpatrick
