Green's Theorem and the Planimeter

Part 3: Calculating Areas in the Plane

1. Evaluate each of the line integrals

   ,   ,  

   over each of the following curves.

   Record your results in your worksheet.

2. What similarities do you see in the values of the line integrals for each curve? How can you account for these similarities?

3. Consider the regions enclosed by each of the curves defined above. What property of these regions appears to be calculated by the line integrals that you have evaluated? Summarize what you see as a conjecture about the line integrals around C and some property of the region D enclosed by C.

4. Define several curves that enclose regions of the plane. Test your conjecture with these curves. Do you need to modify your conjecture in any way or restrict the types of curves to which your conjecture applies?

5. Use your conjecture to find an expression for the area enclosed by the curve x²/a² + y²/b² = 1, using a line integral.
   

6. Write a mathematical argument that establishes the validity of your conjecture.