WEBVTT
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So if we want to evaluate this line integral here
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, one thing that'll be useful for us to remember
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is that this DS here is actually equal to the
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square root of ex prime squared. Plus why Prime
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Squared plus z prime squared. Uh, DT.
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And if we do this, then we could just
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make the substitution for X, y and Z as
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well as, um plugging in N ds for that
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. And then we could just evaluate from 0 to
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2 pi. So let's go out and find out
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what our partials we're going to be. So let
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me actually move the silver. So if we take
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the derivative of this prospective t that just gets his
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ex prime is one, um, why prime is
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going to be? It looks like negative, too
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. Sign to t. And then Z is going
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to be, er ze prime. It is going
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to be to co sign of two teeth. And
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so now if we take these and plug them in
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down here, that's going to be one plus eso
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would be four. Sign to t for co sign
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to t so for signs Square two teeth plus four
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co sign squared two teeth DT. And now if
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we factor that for all we have sine squared plus
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coastline squared, which is just one. So overall
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, this just turns into Route five DT because we
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have one plus four. Okay, so now we
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go ahead and actually let me skip this over and
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I'll move this down here. Yeah, so now
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our bounds are gonna be zero The two pi x
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was t So this is gonna be t squared.
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Why was supposed to be chosen to tea, so
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I'll be cosine squared to t, um, sign
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square to t breezy squared. And then all of
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this was supposed to be multiplied by five DT.
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So we have DT. And I'll just move the
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route five up from here. And so again,
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co sign square plus sine squared is just one.
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So this will think there's one. So we have
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t's word plus one so they integrate both of those
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. We just use power rule. So be route
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five times one third take. Plus, he evaluated
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from 0 to 2 pi. Yeah, so we
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go ahead and plug into pie. So I give
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us Route five and that would be eight pi cubed
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over three plus two pi. Um and then we
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would do minus when we plug desirable. That just
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gives us zero. And so, actually, let
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me just erase this. And then I would just
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say the final answer like this. I mean,
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you could go ahead and get a common denominator on
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all that, but I think this looks fine enough
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.