Parametric Representation for a Contour Integral

Hi, I am having a lot of trouble understanding the concept of "parametric representation" in general. I have the following quiestion,

For the functions f and contours C, use parametric representations for C or legs of C to evaluate the contour integral of f(z)dz.

f(z) is defined by the equations:

{

1 when y < 0

4y when y >0

}

and C is the arc from z = -1 - i to z = 1 + i along the curve y = x^3.

I have the solution for this question, but I would much rather understand the intuition into these types of problems. From my vague understanding, (-1 -i) to (1 + i) is where we would like to integrate from and we would like to use y = x^3 to paramertize f(z).

Am I on the right track? Any help would be very much appreciated.

Thanks