Dave4Math / Courses / Multiple Integrals

Multiple Integrals

Master double integrals and triple integrals. In this course, you'll learn all about multiple integrals in various coordinate systems, and you'll become skilled at making a change of variables in multiple integrals.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Did you know that teacher enthusiasm can lead to better teaching and better student performance? Introducing Dave’s online math course: Multiple Integrals. You’ll see that Dave’s enthusiasm is contagious, and your confidence will increase with each lesson.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in multiple integrals.
  • Students who are enrolled in calculus 3 and want to improve their grade.
  • Anyone interested in getting prepared for linear algebra.
  • Anyone wanting to learn about double integrals, iterated integrals, triple integrals, integrals with a change of variables, and the Jacobian.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Multiple Integrals

  • An Introductory Example
  • Double Integrals Over General Regions
  • Volume of a Solid Between a Surface and a Rectangle
  • The Double Integral Over a Rectangular Region
  • Properties of Double Integrals
  • Iterated Integrals Over Rectangular Regions
  • Fubini’s Theorem for Rectangular Regions
  • Iterated Integrals Over Nonrectangular Regions
  • Polar Rectangles
  • Double Integrals Over Polar Rectangles
  • Double Integrals Over General Regions
  • Mass of a Lamina
  • Moments and Center of Mass of a Lamina
  • Moments of Inertia
  • Radius of Gyration of a Lamina
  • Area of a Surface
  • Area of Surfaces with Equations
  • Triple Integrals Over a Rectangular Box
  • Triple Integrals Over General Bounded Regions in Space
  • Evaluating Triple Integrals Over General Regions
  • Volume, Mass, Center of Mass, and Moments of Inertia
  • Triple Integrals in Cylindrical Coordinates
  • Triple Integrals in Spherical Coordinates
  • Transformations
  • Change of Variables in Double Integrals
  • Change of Variables in Triple Integrals

Course Description

We begin this course with a thorough introduction to double integrals. This approach closely resembles the definite integral from calculus 1. We then discuss double integrals over rectangular regions and work through several examples. At the end of this first lesson, we scrutinize the properties of the double integral.

As we have seen in the first lesson, evaluating double integrals can be cumbersome. So, in the next lesson, we study iterated integrals. We explain Fubini’s Theorem for rectangular and nonrectangular regions and demonstrate when this theorem fails.

Double integrals in polar coordinates are then studied as well as some applications of double integrals. We examine the mass, the moments and center of mass, and the gyration radius of a lamina. We also apply our new knowledge of double integrals to finding surface area, including parametrically.

Next, we begin to explore triple integrals. We motivate their meaning, formalize their definition, and work through examples. We generalize Fubini’s theorem and work through some of the applications concerning lamina again, but with an additional variable. We also discuss probability density functions.

As we studied double integrals in polar coordinates, we now learn triple integrals in cylindrical and spherical coordinates. For both cases, we introduce the coordinate system and then work through several examples. In the exciting conclusion to this course, we explore changing the variables of integration in multiple integrals. This technique involves producing a coordinate system. The strategies and processes involved are detailed.

Recommended Prerequisites for Multiple Integrals

I recommend the prerequisite course Functions of Several Variables. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 3 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.