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Problem 49 Easy Difficulty

Line integrals of vector fields on closed curves Evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ for the following vector fields and closed oriented curves $C$ by parameterizing C. If the integral is not zero, give an explanation.
$\mathbf{F}=\langle x, y, z\rangle ; C: \mathbf{r}(t)=\langle\cos t, \sin t, 2\rangle,$ for $0 \leq t \leq 2 \pi$




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Video Transcript

So in this question where this question were given to field F, which is just X. Come on. Why Commas E and were asked to determine the integral to close integral a f l a g r So we're gonna determine the line integral. And our curve is parameter rised by ex being co sign T. Why being signed T and Z B two and T varying from zero to play. All right, so we're gonna determine after out er while Vester sip. First of all, we're gonna repair amateur eyes f So instead of X, we're gonna put co scientist instead of why we're gonna put society instead of a Z. We're gonna put too. So we get co sign t scientific too. And then now instead of the are we're gonna put our prime DT are our prime is just a derivative of far the derivative of co sign is negative. Sign the derivative off scientists co sign in the derivative off to his deal. Now we just take the dog products are gonna multiply co sign key by negative sign t So we get negative. Scientist ko 70 scientific terms co scientist decide t cost 90 and then zero times two is just so that just vanished. So we get negative Sign t co sign T plus scientific society, which is simply zero. So are lining to grow, be evaluated to be zero.

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