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Chapter9Line Integrals

Objectives

This unit covers the following ideas.

  1. Describe how to integrate a function along a curve. Use line integrals to find the area of a sheet of metal with height \(z=f(x,y)\) above a curve \(\vec r(t)=\left(x,y\right)\) and the average value of a function along a curve.

  2. Find the following geometric properties of a curve: centroid, mass, center of mass, inertia, and radii of gyration.

  3. Compute the work (flow, circulation) and flux of a vector field along and across piecewise smooth curves.

  4. Determine if a field is a gradient field (hence conservative), and use the fundamental theorem of line integrals to simplify work calculations.

Table 1 contains a summary of the key ideas for this chapter.

Surface Area \(\sigma = \int_C d\sigma=\int_C f ds = \int_a^b f \left|\frac{d\vec r}{dt}\right|dt\)
Average Value \(\bar f = \frac{\int f ds}{\int ds}\)
Work, Flow, Circulation \(W=\int_C d\text{ Work } = \int_C (\vec F\cdot \vec T) ds = \int_C \vec F\cdot d\vec r = \int_C Mdx+Ndy\)
Flux \(\text{ Flux } = \int_C d\text{ Flux } = \int_C \vec F\cdot \vec n ds = \oint_C Mdy-Ndx\)
Mass \(m=\int_C dm = \int_C \delta ds\)
Centroid \(\left(\bar x,\bar y,\bar z\right) =\left(\frac{\int x ds}{\int_C ds},\frac{\int y ds}{\int_C ds},\frac{\int z ds}{\int_C ds}\right)\)
Center of Mass \(\left(\bar x,\bar y,\bar z\right) =\left(\frac{\int x dm}{\int_C dm},\frac{\int y dm}{\int_C dm},\frac{\int z dm}{\int_C dm}\right)\)
Fund. Thm of Line Int. \(f(B)-f(A)=\int_C \vec \nabla f \cdot d\vec r\)
Table9.0.1A summary of the ideas in this unit.

Dr. Ben Woodruff has created a YouTube playlist to go along with this chapter. Each video is about 4-6 minutes long.

You'll also find the following links to Sage can help you speed up your time spent on homework. Dr. Woodruff would like to thank Dr. Jason Grout (formerly) at Drake University for contributing many of these (as well as being a constant help with editing, rewriting, and giving me great feedback).

Wrap Up

You've finished the chapter! Look at the objectives at the beginning of the chapter. Can you now do all the things you were promised?

Review Guide Creation

Your assignment: organize what you've learned into a small collection of examples that illustrates the key concepts. I'll call this your chapter review guide. I'll provide you with a template which includes the chapter's key concepts from the objectives at the beginning. Once you finish your review guide, scan it into a PDF document (use any scanner on campus or photo software) and upload it to Gradescope.

As you create this review guide, consider the following:

  • Before each Celebration of Knowledge we will devote a class period to review. With well created lesson plans, you will have 4-8 pages(for 2-4 Chapters) to review for each, instead of 50-100 problems.

  • Think ahead 2-5 years. If you make these lesson plans correctly, you'll be able to look back at your lesson plans for this semester. In about 20-25 pages, you can have the entire course summarized and easy for you to recall.