What similarities do you see in
the values of the line integrals for each curve? How can you account for these
similarities?
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.
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?
Use your conjecture to find an
expression for the area enclosed by the curve x^{2}/a^{2 }+ y^{2}/b^{2 }= 1,
using a line integral.
Write a mathematical argument
that establishes the validity of your conjecture.