Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path $C_{1}$ followed by $C_{2}$ followed by $C_{3}$ from (0,0,0) to (1,1,1) (see accompanying figure) given by

$$C_{1}: \quad \mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1

$$$C_{2}: \quad \mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$$C_{3}: \quad \mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$

(GRAPH CANT COPY)

## Discussion

## Video Transcript

Okay, folks. So in this video, we have this line into grow, which is the square root was just the integral against the square root of X squared plus y squared, Yes, along the curve. See? Of course. So the way we're the way we're gonna evaluate this London to grow is the usual way of evaluating of light of exceeds member free states. The way we're gonna do this line in the grow is by first converting the DS into something Times d t o k e. I mean, this is all that's just the usual trick. The re privatization is variables with respect to another variable and all that kind of stuff. So we have d s. Usually DS can be written as ex prime squared plus y prime squared plus z prime squared The whole thing multiplied by DT. I'm not gonna go over why this is true. We should know that why this is true. Um, so now we have what's ex prime ex prime? It is Thea time derivative of X, with respect to t um X, as a function of T is for co Scient E because it's given in the problem. So we have, um four of minus signs. He because Because you take the derivative of co sign with respect to the variable. You get negative sign this thing squared. Plus why Prime is just why co sign of t squared plus Z is a function of teach three t. So when you take the derivative of that, you get three squared, multiply by d t. And now you have 16 sine squared 50 plus co science grade of T. I'm sure you know that this is just one 80. And so we have this one here we have 16 plus nine, which is 25 d t. Which is gonna give you the square root of 25 which is 55 DT. So the efforts five DT and now we're gonna plug in this into the integral, which is gonna give you Okay, so let's see the integral The Inter Grand is the square root of X squared plus y squared. OK, but the square root of X, where plus y squared I'm gonna jump a few steps here. I'm not gonna go over every step but the square root of X squared plus y squared just 16 times CO sine squared plus sine square, which is one that's that's the square root of X squared plus y squared. OK, so the S is five d t. I pulled the constant five up in front. I can do that because that's one of the properties of integral, which is that you can pull and a constant from inside than the integral outside it. Integral. Okay, so we have five times the square root of 16 which is for five times four. Um, integral. With the tea, we have 2020 times the integral of the T. The integral of one time stay. Tea is just tea evaluated at the two limits. The first limit, the bottom limited. Negative two pi, apparently, for some reason. And the top limit is two pi. Okay, so we have 20 of two pi minus minus two pi, which is 20 of two pipelines to bite is four pi, which is 80 pie. That's a weird, funny number. Okay, we're done for this video. Thank you.

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