how do you find the area under this space curve or any other space curve equation: r(t)=(4*t^2)I+(1/t)J+(-2*t)K
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: $\displaystyle \int_{t_0}^{t_1} z(t)\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2}dt$