I have no idea what you mean by "area under a space curve". Do you mean "the area swept out by the line from (x, y, f(x,y)) to (x, y, 0)" where z= f(x,y) is the curve? And what are the minimum and maximum values for t?

If that is what you mean then the area is given by the integral of z with respect to the arclength of the curve projected into the xy-plane: