That's what I got from the non-Stokes approach:
∫F dot c'(t) dt from [0,2pi] = -16pi
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Not sure if you've used Wolfram Alpha before, but it serves as a great check on your integration. This particular integral was a bit, erm, long for the free service, but the answer -50.26...
Looks correct to me. The integral isn't as evil as it appears. I tried to integrate the four final terms separately... For three of the terms, it seems that with a bit of w-substitution, you should be able to redefine the bounds to a form where you won't need to evaluate the integral at all. An...