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Preview
Preview
This lesson begins our study of multiple integrals. We’ll focus on double and triple integrals over rectangular regions in this lesson.
In Module 1 we’ll start by learning how to approximate the “net signed volume” between the graph of a two- variable function and the plane using Riemann sums (analogous to how we arrived at single-variable definite integrals by investigating how to measure the “net signed area” between the graph of a function f(x) and the x-axis). We’ll then define double integrals from this and learn simple techniques for evaluating double integrals when the region of integration is a rectangle.
In Module 2 we’ll redo all this work for triple integrals, again learning how to evaluate them in the simplest case—when the region of integration is a box in 3D.
Finally, in Module 3 we’ll discuss a few properties of integration that are analogous to the properties we already know from single-variable integration.
Review
Review
Integration from single-variable calculus: Paul's Online Notes contains a good review.
Learn
Learn
Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson Notes
Lesson 10.pdfModule 1 Video
Module 1 Video
Module 2 Video
Module 2 Video
Module 3 Video
Module 3 Video
Reflect
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 10 PP.pdfPage updated
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