Direction of Contour Integration When I'm using the residue theorem to evaluate a contour integral, does the simply closed curve always have to be in a counter-clockwise direction? I believe that I can go in a clockwise direction, but this adds a negative sign, correct?

Reversing the orientation changes the sign of the result. This is visible when you change the contour integral to a normal integral: $$\oint_{\gamma}f(z)dz=\int^b_af(z(t))z^{\prime}(t)dt=-\int_b^af(z(t))z^{\prime}(t)dt$$ This can be done with higher dimensions, too.