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Problem 8 Medium Difficulty

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x^{2} d x+y^{2} d y, \quad C \text { consists of the arc of the circle }} \\ {x^{2}+y^{2}=4 \text { from }(2,0) \text { to }(0,2) \text { followed by the line }} \\ {\text { segment from }(0,2) \text { to }(4,3)}\end{array}$$

Answer

83$/ 3$

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Video Transcript

So in this question we're giving the were given. You were asked to find the line integral of X squared D explosive voice, do I? And we're given that the curve conscious of two parts. First, it consists of the Ark of the Circle X squared plus vice, where people sport. And that goes from zero to you too. And the second part consists of the, uh, lying segment, which goes from zero to what if we draw this on a graph? So see, one just goes from zero, uh, zero too young to, And it's a circle of radius, too. So 1/4 of a circle of radius, too. And then we're given the sea to simply a line segment. Okay, so says just buying second like this. All right, so first of all, we know how to pair of parameter rise circles, and we noted for amateurs like, So first, we're gonna start with C one. So are C one is on the quarter, the circle. So again, since it's a circle of radius for, uh, Well, when we write it when with a rabbit, arise it excess to close. I Inti lies to sign t and then our, uh are tee. Our TV goes from zero to pi over two. All right, so here it's clear. We started syrah. Go all the way up department. Okay, good. And then now, uh, we're gonna find the XO. The X is just the derivative of X, with respect it easy access to society. If we take the derivative of that with respected, you get negative to sign t and again, do you? Why the derivative of why with respected tea instead to coast 19 Right now, we're gonna plug everything back into the integral, and everything is gonna be in terms of tea. So are Lower LTD's from zero in our upper limited spy or two. And then we plug everything back. So place of expert enough to close 90 in place of why we're gonna two scientists in place of the expert gonna put negative to science t t in place of why we're gonna put do coastline. So we plug everything back. And then after we simplify, we get something, we get this right over. All right. Now, what can we pull out as a common factor? Well, there are multiple things we can float is a common factor. We're gonna pull out in eight, and then we're just going to switch. He's like this because this has a minus. I don't want The money is to be so Now are integral looks, Chris. All right, now let's seek This is a bit tricky because we're gonna have to do a double U substitution. So for the first part right over here, we're gonna set you equal sign, tear. And we have the EU is close 92. And for the second part, we're going to sit w equals two co society, and we get deep W is negative. Scientific et again, this is just a double use institutional tree already hard. Well, we should We should not tell right now. All right, so now we think the integral if you swear to you, that's just you cute, give you. And then we know that you in this side of you. So we get this treachery. And now the integral W Square. You tell your masters W Cube divided vice, and we know that w is co sign tease right over here. All right. And there's an eight on the outside. And remember, this is a definite integral. So we're going from pi over from zero to pi over two. Certainly plug this in by over two and zero and you subtract. We're going to get eight times 1/3 0 which is just one. Mine is your A plus 1/3 which is also so we have eight times 1/3 minus one 1/3 minus prefer to zero 1,000,000,008 times zero, which is just okay. Now it's looking at C two. So see to we get the integral of Bill C to we have to parameter rise the line. So if we parameter rise the line, we know first, we need to find the direction in which the line is pulling. So we do X to marina sex sandwiches for minors, Europe is just four. And then why? Till minus final in which is three minus two, which is one. So then we get our parameter arised, uh, are parallel line is for so the line segment zero to the line for line segment connecting 0 to 43 has to be parallel to for all right, and then now we determine X and one So X is just explain, plus a teeth for X one is zero plus a is war. So zero plus 14 just 14 is our ex. Our Why is why one plus Bt's All I want is to and he is wrong. So it's two plus one team and our t goes from 0 to 1. Why, with bunions Europe 14 get X equals zero. Why you two That's our first point. They plug in 1 40 we get X equals four y equals three, which is our second. And then again we're going to determine DX and dy y. The X is just a derivative before TV respected team just just 40 and the wise a derivative off two plus one to you and respected t just just tt okay, Now we plug everything back into the integral look everything and when we get everything in terms of but our limit points are from 0 to 1. Okay, so now if we expand if we foil this right here and we multiply the side weekend 60 40 squared plus four plus support plus two squared. Now again, this we can combine the 60 40 squared. With these squared, they get 65 squares of get this integral over here, and this is not difficult. This is just the power room. So here we're just gonna integrates with the integral of 65 the spirit of 65 Cube, divided by three. Integral of forties to the square an angel for is just tea and are integral goes from 01 So if we plug in one, we get 65/3 plus two plus four minus of plug ins. 0 40 Ridge is gonna get zero. So it's 80. So now we do the we look right over here, and then we create a common denominator. We're going to get that the line integral is 83 divided by three. Now, if we add a result for the first line integral for the quarter circle and then re at our second result which was the line integral for the line segment, we're gonna get zero plus 83 divided by three. It is just 83 divided

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