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Emkay.C E.

Calculus 3

1 month, 2 weeks ago

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Find the length of the curve. $$\vec{r}(t)=2 t \vec{i}+\ln t \vec{j}+t^{2} \vec{k} \text { for } 1 \leq t \leq 2$$

Chapter 17

Parameterization and Vector Fields

Section 2

Motion, Velocity, and Acceleration

Hello there. So for this exercise we need to calculate the length of the curve on this interval between one and two. The curve is given by two T. In the component. L N F T. At the jade component. Anti square indicated component. So let's remember the the length of a curve on a given interval is calculated by taking the integral between one and two between the The interval of interest that in this case is between one and 2 of the norm of the first derivative of the curve with respect to T. So we need to calculate first the first derivative of our curve. So the first derivative of our curve is going to be too in the component plus one over T. In the J. F. Component and plus two T. In the J indicate come point but we need the norm of this vector. So the norm of this picture of the first derivative is equal to the square root of four Plus one over the square plus for T. S. Square. And we need to replace this into our integral. Okay, so from that we obtain an integral between one and 2 of uh T and here in the square root we're going to put eight T. S. Square plus one. The um sorry sorry I have a mistake here is four T. S square plus four T. To the four power plus one. Yeah, that's it. Yeah. So we have this integral here. We can solve this by parts if I'm not wrong. Okay. Um No. Okay look so this yeah, so this is going to be equal to the integral within one and two. And this part inside here is equal to look to the square root of to T. Plus one square over T. Meeting. We can eliminate here the square root. So we obtain a simpler integral within one and two off to T. Plus one Divided T. Then team. Right. Uh This I'm going to regret this here. So the length is going to be equal to the integral through one and two of two plus one over T. VT. And there's a result into taking two times the integral Between two and 1 and plus the integral of one over T. Bt. This. Okay. Um Yeah so here we have that this is equal to two plus the law. The island of T Between two and 1 and there's equals to two plus the end of two minus the end of one. But Ln of one is equal to zero. So we don't take into account this and this is the length of the skirt

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