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Green's Theorem and the Planimeter

Part 6: Summary

  1. Write down a procedure describing your method for calculating the line integral $\int_{C} P(x,y) \; dx + Q(x,y) \; dy$ directly (i.e., without resort to iterated integrals).

  2. How can iterated integrals be used to evaluate line integrals? What opportunities does this relationship between line integrals and iterated integrals present?

  3. Suppose you had a region of the plane that resembled the region D shown below. How could you establish whether or not the relationship between line integrals and iterated integrals worked for such a region?

  4. Explain how the relationship between line integrals and iterated integrals can be used to calculate areas of the plane using line integrals.

  5. Suppose you were given a planimeter that had no information about how to convert the readings from the planimeter into areas. How would you go about establishing the relationship between the readings from the planimeter and the corresponding areas?

 

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