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THE GRID
Essentials
Main ideas
- Understanding different ways of expressing area using integration.
- Concrete example of Area Corollary to Green's/Stokes' Theorem.
We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.
Prerequisites
- Familiarity with line integrals.
- Green's Theorem is not a prerequisite!
Warmup
- The first problem is a good warmup.
Props
- whiteboards and pens
- a planimeter if available
Wrapup
- Emphasize the magic – finding area by walking around the boundary!
- Point out that this works for any closed curve, not just the rectangular regions considered here.
- Demonstrate or describe a planimeter, used for instance to measure the area of a region on a map by tracing the boundary.
Details
In the Classroom
- Make sure students use a consistent orientation on their path.
- Make sure students explicitly include all segments of their path, including those which obviously yield zero.
- Students in a given group should all use the same curve.
- Students should be discouraged from drawing a curve whose longest side is along a coordinate axis.
- Students may need to be reminded that $\OINT$ implies the counterclockwise orientation. But it doesn't matter what orientation students use so long as they are consistent!
- A geometric argument that the orientation should be reversed when interchanging $x$ and $y$ is to rotate the $xy$-plane about the line $y=x$. (This explains the minus sign in Green's Theorem.)
- Students may not have seen line integrals of this form (see below).
Subsidiary ideas
- Orientation of closed paths.
- Line integrals of the form $\INT P\,dx+Q\,dy$.
We do not discuss such integrals in class! Integrals of this form
almost always arise in applications as $\INT\FF\cdot d\rr$.
Homework
(none yet)
Essay questions
(none yet)
Enrichment
- Write down Green's Theorem.
- Go to 3 dimensions — bend the curve out of the plane and stretch the region like a butterfly net or rubber sheet. This is the setting for Stokes' Theorem!