Explore a comprehensive series of video lectures on multiple integration, covering double and triple integrals, coordinate systems, and change of variables. Learn to define and evaluate double integrals, apply Fubini's theorem, and integrate over general regions. Master integration techniques in polar, cylindrical, and spherical coordinates. Dive into practical examples, including volume calculations for various geometric shapes. Understand the Jacobian and its role in coordinate transformations. Enhance your understanding of advanced calculus concepts through clear explanations and diverse problem-solving approaches.
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
