A curve is formed by the intersection of y^2/9 + z^2/4 = 1 and the plane x-2y-3z = 0. The particle moving along the curve goes from (6,0,2) to (-6,0,-2). Find the work done on it by the vector field F(x,y,z) = -yi + xj + yzk.
I'm going to need to find the integral of F dr which is actually going to be the integral of Fr(t) dot with r'(t) dt.
The Attempt at a Solution
I just need to set the integral up correctly, because I'm going to be letting the computer do the integration. So...
My main stumping point is coming up with the correct parameterization for r(t). In order to find a curve formed by the intersection, I used 3costj + 2sintk and then, for my i component, I noted that the plane equation gave me x = 2y + 3z. I then stuck in the y and z components I had (3cost and 2 sint, respectively) and came up with this curve: (6cost+6sint)i + 3costj + 2sintk. That's where I come to an abrupt stop...because I need r(t) going from (6,0,2) to (-6,0,-2), and I'm not quite sure how to do this. More than likely simple, but I'm stumped.
I know that once I get an r(t), I need to stick it into my function F, and then dot itself with the derivative of r(t). I don't think that will be that difficult, especially if I can simplify some (for instance, I see I could simplify the i component of my hpothetical curve to 6(cost+sint)i.
Any assistance would be much appreciated!