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

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

Compute $\displaystyle \oint (e^{x^2} - 2xy)dx + (2xy - x^2)dy$.

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

Compute $\displaystyle \oint (4x^3y - 3y^2)dx + (x^4 + e^{\sin(y)})dy$.

For (a), my method was to split the curve in five curves, $\displaystyle C_1 \to C_5$, where

$\displaystyle C_1$ goes from (-1,1) to (3,1),

$\displaystyle C_2$ goes from (3,1) to (4,2),

$\displaystyle C_3$ goes from (4,2) to (3,3),

$\displaystyle C_4$ goes from (3,3) to (-1,3), and

$\displaystyle C_5$ goes from (-1,3) to (-1,1).

Then I parametrized each of the curves as follows (where $\displaystyle 0 \leq t \leq 1$ for each curve):

$\displaystyle C_1: x = 4t - 1, y = 1, dx = 4dt, dy = 0$,

$\displaystyle C_2: x = t + 3, y = t + 1, dx = dt, dy = dt$,

$\displaystyle C_3: x = 4 - t, y = t + 2, dx = -dt, dy = dt$,

$\displaystyle C_4: x = 3 - 4t, y = 3, dx = -4dt, dy = 0$, and

$\displaystyle C_5: x = -1, y = 3 - 2t, dx = 0, dy = -2dt$.

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 $\displaystyle e^{x^2}$ 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.