WEBVTT
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All right. So this question we're asked for determined
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the line integral of back stands. Why? Times
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either the power of y z along the curve giving
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Perry metric Lee by access, equality wisely, quantity
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squared been C is equal to t cube Anti goes
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from Syria. Well, the first thing we want
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to do is we're gonna We want to write everything
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in the integral in terms of teeth while exes,
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Exes in terms of TV wise in terms of But
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our problem is that de y is not in terms
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of so what we can do is we can take
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the derivative of eyes equal to t swear. So
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you wise just gonna be to tee times. DT
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. All right, so now we're gonna plug everything
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back into our starting a plug in for X t
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. Where weren t squared for TV for a CD
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, cured a t to the power of three.
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And for d y, we're gonna plug in to
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Tgt. So my plug everything into the integral we
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get. We can pull this to to the outside
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, Take it to the outside. We get two
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times the integral from 01 cheated the power for times
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e to the power of to the power five.
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All right, now what we can do is we
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have to solve this. Integral. This looks like
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a product, but we're not gonna use integration by
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parts. What we can do is we notice that
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there's a teacher about five on the NASA inside the
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exponents. So we're gonna get you equal tedx in
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a power. All right? Now, when we
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take a derivative both sides, right, I d
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year is equal to five. T to the power
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40 t and then we can divide by five on
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both sides, and we finally get that t to
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the power or GT is equal to the U divided
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by five. So this and this is do you
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divided by five and then we have to. Then
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then then we have e to the power of you
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. So basically, we could pull this 1/5 of
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the outside, so you have to hit integral you
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and then we can change the limits off our integration
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so that they can be in terms of you.
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So we just pull against a T is equal to
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zero. If we pluck that and we get zero
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to the power five, which is just zero.
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And again T is equal to one of the one
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change that you just becomes you is equal to one
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to the power of five or just one. So
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now we change their limits of integration. And we
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know that the integral U to the power of you
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use just beat of the power of you. So
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we have to fifth and now we they eat it
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the power of one minus eight to the power of
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zero. But we know that either the power of
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zero is so eight on the power of zero is
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one. So our final answer the line integral is
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equal to two e minus.