• 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.

• Familiarity with line integrals.
• Green's Theorem is not a prerequisite!

• The first problem is a good warmup.

• whiteboards and pens
• a planimeter if available

• 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.

• 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).

• 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 $\INT\FF\cdot d\rr$.

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• 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!