Problem: Use symmetry considerations to find the following. (Picture of curves attached.)

(a) Let C be the polygonal curve shown in Diagram (a).

Compute .

(b) Let C be the curve shown in Diagram (b); you might visualize as a racetrack with two semicircular ends.

Compute .

For (a), my method was to split the curve in five curves, , where

goes from (-1,1) to (3,1),

goes from (3,1) to (4,2),

goes from (4,2) to (3,3),

goes from (3,3) to (-1,3), and

goes from (-1,3) to (-1,1).

Then I parametrized each of the curves as follows (where for each curve):

,

,

,

, and

.

Then the next step would be to split the integral into five integrals, one for each curve, and add them together. But the problem is the term that hinders my integration attempts after the parametrization, and I can't evaluate the integral.

I assume that there must be a better approach to doing this problem since I haven't actually used symmetry as it asks. But I'm not exactly sure how to apply symmetry to curve integrals.