Evaluate $\int_{C} \sqrt{x+2 y} d s,$ where $C$ is

a. the straight-line segment $x=t, y=4 t,$ from (0,0) to (1,4).

b. $C_{1} \cup C_{2} ; C_{1}$ is the line segment from (0,0) to $(1,0),$ and $C_{2}$ is the line segment from (1,0) to (1,2).

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Okay, people. So in this video, we're going to take a look at problem number 20 where we have a line integral to solve on. And there's two parts of this we have part in. We have part B. So, um, so for part A. Well, let's do party first. Um, for part A, we have weren't have to do this. We have X as a function of t is simply equal t. And why, as a function of T is simply equal to 40 So we're going to be doing this line and a girl of C um, from X Plus two y but I'm gonna rewrite X as a function of T in M. And same thing with why I'm gonna rewrite y is a function of t. So I'm gonna plug in t here for for X. So I have X, um equal, steep plus two times why? But why is 40 so two times that equals 80 multiplied by the Yes, if you remember what the yeses ds is really just the square root of, um ex prime squared. Plus why Prime Squared multiplied by DT. Um, and of course, I'm gonna substitute in ex prime squared, which is once word, which is just one, um, and then I'm gonna substitute in. Why Prime squared? Well, wise 40. So why prime? It's four. Therefore, four squared is really just 16. So I have I'm gonna pull out a constant here, which is the square root of 17. I pull this thing out in front of the integral, and then I have Ah, the square root of 90 d t left. Let's figure out what the limits of integration are. Um, Well, we started from this is our beginning. Point are beginning point has 00 Um well, xz with a t when x zero. That means t is also zero. So that's my, uh, the first limited integration. What about the top? Top of it? Well, for for the endpoint here we have X equals toe one. Well, because of the fact that ecstasy with a t x equals y means that t 01 So this is my top limit of integration for tea. So let's figure out what this integral is what we have skirt of 17 times three because there is a spirit of nothing here which I pulled out of the integral um, And what I have inside it in tomorrow is this. I have t to the power 1/2 DT. Um And so now I have three Route 17 t to the power 3/2 over 3/2 from 0 to 1. And I'm gonna cancer are gonna cancel out this three year. So now I have to route 17 times. One. Um, my zero. Of course. Which I'm going to ignore. Okay, so this is the answer for party of the problem that's start doing part B. Well, Barbie is a little bit less trivial because we have We have really two curves here we have. This is C one C one does the curve from X equals zero all the way to X equals one. And C two is the curve. Um, it was not really a curve is the line segment, but you get what I mean. Um see twos. The segment from from why equals want from white was dearer to zero all the way up. The white was too. So, um, we're not We're not really given. Were not really parameter rising. If you notice you look at the problem here, um were not really parameter rising X and Y with with another variable tea or whatever were just given x and Y and were asked to evaluate this London to grow. Therefore, we're gonna have to utilize what we know from this graph. Okay, So if you let's take a look at this integral here we have integral long sea of square root of X plus two. You Why, yes, we have the same tomorrow here, but we have to curves that's going to make our life a lot difficult, A lot more difficult if we don't. If we don't like somehow separate this one, um, curve, see into two parts. So we'll have to separate this. So integrating along sees the same thing as integrating along C one plus integrating along. See, too, because that's the same thing because C is C one joining joined with C to there. That's the same thing. Um, so for the first integral, which is a long sea one, if you notice if you look at this graph here everywhere along this line segment see one why the value of why is fixed and that value is zero. So so when you when you plug that value of zero. In here, you get zero plus X. Um, and so act in which I could just rewrite as square root effects. Okay, because there's zero following it, which I'm going to ignore um, multiplied by the Yes, but if you know this, the U. S is the same thing is the same thing as the X. Because for C one, it's really just the line segments he wouldn't. It's just a line segment from 0 to 1, actually, with zero all the way to actually another one. So the S is the same thing as DX because you're not varying. Why? Um, So I think that's, um that's for the first curve. C one. What about the second curveball is really the same logic. The second curve everywhere along the curve c to hear everywhere along the currency to he noticed that the X value is fixed. So I'm gonna existing a substitute, um, the value for X, which is one in here plus two. Why multiplied by the Yes, But if you know this DS is the same thing as the wind, that's the Chinese. Their lives are going to be a lot easier than then. What it was before. So let's now see if we can figure out what the limits of integration is. Okay. For C one, the limits of integration is really a simple it's religious. X equals zero all the way to X equals one. Okay. What about what about the second curve for the the second girl? Well, the second world goes from why equals zero all the way up to why equals two. All right, you could just read that off the graph. All right, so I think that's it. That that's the That's, um that's the strategy. We have successfully converted one chunk, one big integral into two. Little into girls, which weaken separately. Evaluate. So we have exited power. One have the integral of that gives us the power. 3/2 over, 3/2 from zero, the one plus the 2nd 1 Well, the 2nd 1 is a little bit trickier. Um, we're going to do a little bit of ah, you know, you substitution here. We weren't defined you as whatever is inside of the integral. I mean, whatever is inside of the square root, which is one plus two. Why, that means to you is to see Why, great. Um, and then I'm gonna substitute that back in here. We have you to the power three. I mean, to the power of 1/2 multiplied by D Y. But the wise one have, Do you? So we have do you over to, um let's rewrite the limits of integration as, um Well, why is zero you is one when why is to you is one plus for which is five. Um, so let's take care of the first drill. We have two or three times. One might. Zero. Plus, I'm 1/2 of you 3/2 over 3/2 from one device. Okay, so we have to third plus 1/3. I kind of pulled the constant out in front, um, five to the three. Half minus Wonder the 3/2. So let's ah, destroying. That's ah, group these constants. Together we have to third plus 1/3 times five to the power of 3/2 minus 1/3. Okay, so we have to third minus 1/3 gives you 1/3 um, one plus five to the power three halfs. All right. I think, uh, that's it for this. For this problem, we have successfully solved integral for apart and in grow for part B. We haven't done them all. That's it for this video. Um, thank you very much.

University of California, Berkeley