I'm not sure what you are doing! In particular, I have no idea why you are looking at the individual sides of the hexagon. You were told to do this using "Green's theorem" which says that the integral of a vector function, , around a closed path, counterclockwise, is equal to the integral of over the region bounded by the closed path.
Here the vector function given is so that f(x,y)= 0 and [/tex]g(x,y)= x^3[/tex] so you want to integrate over the hexagon. Since that is symmetric about the y- axis, it is sufficient to integrate of x= 0 to and double. The lower boundary is the line from (0, -2) to . The slope of that line is and the equation is . Similarly, the upper boundary is the line from (0, 2) to . The slope of that line is and the equation is .
So the solution is .