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Preview
In single-variable calculus we learned all about integrating a function y = f (x) over an interval [a,b] on the x-axis. But what if instead of integrating f over the line segment [a,b] on the x-axis, we integrate it over some curve in the xy-plane? What we get are called “line integrals,” though I’d prefer a name like “curve integrals,” and in this lesson we’ll learn about them.
In Module 1 we’ll focus on such “line integrals” over planar curves—that is, curves in a plane (like the xy-plane)—that are parameterized by arc length. We’ll learn that these line integrals basically generalize the arc length formula you learned in single-variable calculus.
In Module 2 we’ll expand our study to line integrals over curves parametrized not by arc length. Not much will change, but we will at the end of the lesson tie everything back to vector-valued functions, revealing some neat insights.
Review
Review
Arc length from calculus; this webpage has a good review of the topic.
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 13.pdfModule 1 Video
Module 1 Video
Module 2 Video
Module 2 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 13 PP.pdfPage updated
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