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# Find the work done by the force field $$\textbf{F}(x, y) = x \, \textbf{i} + (y + 2) \, \textbf{j}$$ in moving an object along an arch of the cycloid$\textbf{r}(t) = (t - \sin t) \, \textbf{i} + (1 - \cos t) \, \textbf{j}$ $0 \leqslant t \leqslant 2\pi$

## Work done $=2 \pi^{2}$

Vector Calculus

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work done by the force pills. I want to compute this and ah, that is done by computing. If off our tea which access replaced by T minus I Inti why is replaced by one minus co sign t saw three minus co sign and the R pry off T is one minus co sign t ah, sci fi. So they're thought products or the integral becomes in a goatee from zero to two pi and there's that product So that product should be t minus tico sai inti minus sign t close Scientific society plus three society minus scientific society. So we just cross out this term So this one zero to two pi t minus he cause I he plus two side Did he this home So maybe we'll have to do a little bit integration by part here. So in the growth Pecos I Inti, is you a sad issue this TV So you times you would just be sign He minus the to you is just one one dt so t sign t minus koh sai nt So here the entire derivative on the the entire derivative here should be t square over too, minus thiss minus T sai Inti plus Co sign T and the entire reviews Negative too Cose I t so this one cans are becomes minus sign minus koh sai nt probably to pylea four pi square over too We get to Pi square minus to pie Scientific I zero So we'll have the worry about this time and ah co sign to pie This is the same as cousins who just want and the miners If you probably zero well zero here zero here cause I know this one so minus one so minus should be minus next before So here we have a problem to pi We got this minus republic Probably zero We've got an active one So here we got two Pi square minus one plus one so

Vector Calculus

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