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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.


  • 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.


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 $\INT\FF\cdot d\rr$.


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Essay questions

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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!

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