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Problem 46 Hard Difficulty

If $C$ is a smooth curve given by a vector function $\mathbf{r}(t),$ $a \leqslant t \leqslant b,$ show that $$\int_{C} \mathbf{r} \cdot d \mathbf{r}=\frac{1}{2}\left[|\mathbf{r}(b)|^{2}-|\mathbf{r}(a)|^{2}\right]
$$

Answer

$\int_{C} \mathbf{r} \cdot d \mathbf{r}=\frac{1}{2}\left[|\mathbf{r}(b)|^{2}-|\mathbf{r}(a)|^{2}\right]$

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Video Transcript

So in this video, we're s too. Show that the line integral along the curve off our deal are are dot your is equal to 1/2 the magnitude of bar B square minus the magnitude of our age square as t goes from a to B. Alright, So first of all, our our t r curve or position Victor So we can write that in terms of x, y and Z So it's except e I have much wife t dream had Muzi of t. K. All right now, we can, uh, write this as a vector. So bars just exit t call alive t columnist in your teeth and then are are fine. We just take the derivative bets. So our prime of tea is just experience. Do you call the white crime of commas? All right, Now take the right hand side of the integral along. See our our dog to your we can write that as integral in terms of teas, though it's the integral from eight of the off our t dot R. All right, so we know what our He is just excellently life museum. And we know what our primal is just our privacy. Sorry. Experiment. You like travel to use. Okay, so when that wouldn't be multiply this out. We get integral from a to B. Yes, except E except in Times Express, if you aspire to times like private was zero times the property. All right, so now we need to notice We know that what is accepting expert, will we take the derivative of X square of TV? Well, if we use our changeable so we know that this is a function of the functions we're gonna user change room So it's gonna be d of x square t by the X, and then we're gonna multiply that by D of x of t you right, Because we know that excellent option t So this is gonna be the change will. So this is our chamber. But what's the derivative of X squared with respect, X levels just to apps. And once the exp i t t Well, that's just express So and here we notice we have except e x times expert, but are only issues that we have. This little happen front so exits off. We divide both sides by two. We're going to get x x times X prize is 1/2 that derivative. Well, thanks square. And then we can do the same thing for a while and see, So we're going to get this right over here. So they substitute xx bribe. I would have the derivative Rex square with respected T. And if we substitute why times like prime with 1/2 the derivative before squared, respected Tiu. And if we substitute ze time, see a prime by 1/2 the derivative of deceased square. Then what we can do is we notice that there's a common factor off what happened. So we pull this one after the upset. Remember, If we're gonna take the integral off a derivative by the fundamental theoretical calculus they can't select. So what? We're left with its X squared of going from A to B. Why squared of t going for be a to be and the square of people. Now if we pluck me, is it We're gonna get 1/2 x squared of be Linus X squared away Plus why scared of B minus y spirit of a plus the square to be by the sea script. I'm just rearranging putting the negatives together and depositors together. We're going to get this fine. It's this. I don't know. What do we notice? We notice an expert of people swear spirit of people's the spread of B. Well, that's just the magnitude of B squared. We also noticed I get X were debate plus y squared of any Percy spread of able. That's just the magnitude of Barbie Square. We have 1/2 Barbie outside and that we showed. I want you to be sure 1/2 times thing magnitude of R B squared, minus the magnitude of our school. That's the link.