Find the mass of a wire that lies along the curve $\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, 0 \leq t \leq 1,$ if the density is $\delta=(3 / 2) t$.

$2 \sqrt{2}-1$

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Okay, folks. So we're gonna talk about problem 33 here, Um, we're gonna be looking for the mass of the wire. And as you recall, I just want to remind you that there, when you have an infinite asthma line segment with a certain density delta than Delta multiplied by the S gives you D m. That's the density. I mean, that's the definition of a density. Density is, you know, mass per length. So when you do an integral of this, you're going to get the mass of the of the entire wire. Okay, that's really the idea. So that's let's calculate, um, our problem. Here we have. We have to look for the mass of the wires that we do an integral. We have dealt with the eyes. Delta is gonna be three have tea and a DS can be calculated, Could be written in another way, could be written as four plus weren t squared. Now the way I got this is by realizing that they were given a function for why, as a function of T, which is T square minus one, and were also given a function for Z, um, as a function of T, which is equal to duty and a DS can be written as why Prime Squared. Why Prime is to t z prime is too. So why Prime Squared is 40 squared plus c prime squared once the promise to So that gives me for multiplied by DT So this is just another way to write infinite decimal I incitement Azan expression for team. Okay. And the integration limits is between zero and one. That's what that gives me three t one plus t squared t t. And now I'm gonna do a use substitution, which I'm sure you've all heard of. One place to use squared. That's what I defined to be you. And do you is to t d t. So now I'm going to substitute this back in here. We have three. This is your the one. This is, you know, 2312 you have you over to and that is going to give you two to the power of three. Have minus one. And I think that's it for this video than mass. Whatever unit this is the masses to to the power of 3/2 minus one, which can be rewritten as to um multiplied by route to minus one. However, you want to write it and they're the same thing. OK, thank you for watching.

University of California, Berkeley