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Evaluate the line integral, where $C$ is the given curve.

$$\begin{array}{l}{\int_{C}(x+2 y) d x+x^{2} d y, \quad C \text { consists of line segments }} \\ {\text { from }(0,0) \text { to }(2,1) \text { and } \text { from }(2,1) \text { to }(3,0)}\end{array}$$

$\frac{5}{2}$

Vector Calculus

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Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

All right, so in this question, were asked to determine the group off. Experts still idea X plus X squared D y. And we're given that the curve consists of two line segments one going from 002 to 1 and the other blowing from 212 threes. All right, so the first thing we're gonna do is Bergen and parameter rise are curve. So we have to. Kurds were going toe parameter. It's both. So the first thing we have to do is we're given 00 to 1. We need to find the the factor that said parallel to the line connecting these two. So what we do is we subtract x two x one and why to from why one so we get to one. So this is our a This is B And then again, where do you know how to do this? So its X one plus 80 I will give us act so excellent zero plus a is two times t so access just to t Same thing Why is equal to y one plus Bt by 10 and he is one and t s t. So we just get that y is equal to one times t so we should already like we already know this from before the perimeter trees and again our r t is well, those from 0 to 1. Because when he is equal to zero, we get X equals zero equals zero. And when tea cools one, we get two times on the two and to one. So this is exactly what we need. We need are are lying to both of 002 to 1 and it does that. And now I'm just gonna put a little I'm gonna add something extra that might not make sense, but it will, reader, we're gonna find the X. And so we know that our access to T so DX is just to DT, right? The same thing Why is won t so d lines just DT So we determine the expanded. All right, great. So right now we're gonna goto our original, um, line into group right here, and we're gonna plug in our X or why are dx dy y? And we want everything to be in terms of tea. So again, our limit points are going to be zero and one because he goes from 01 We're gonna plug everything back in and everything should be in terms of We're gonna forget her acts or lie r d X and the y Everything's gonna be in terms of teeth. So now, after we duel the algebra, the dust settles, We get that we need to find the Inter Group of 01 0 18 plus 40 square DT. All right, so what we did is we distribute this too. So we get yeah, and then also we have, Yes. So that's a teeth. And then here to t swear it is just working to t all that squared. It's 40 square, and then we take the integral of that. So we get 80 square divided by two plus 40 killed, divided by three, and then our limits of integration off 01 So if I plug in one, we get four plus 4/3 minus. If it looking zero, we're just getting so there are four plus for over three. Another way to write for is 12 to but by three. Now we have a common denominator. So this is just 16/3. So now we determine the integral or the line segment That was from 002 to 1. Now, again, we need to find We need to find a line into grow for the segment that goes for 21230 All right, again, we need to determine the direction off the vector A vector parallel toe to toe line. So we're just gonna subtract x one x two of x one And why to for my one. So we get three minus two, Call a zero minus one. So we get one common negative one, then our Formula two pair amateur eyes, The exit allies just x is X one plus 80. So this is a This is be so x one is two plus So this is X one Sorry x one. Why one? So this will be two plus eighties or two plus warranty. And why is why one waas bt So I want it's one use negative one and teaser. It's what dynasty again. Our t goes from 0 to 1. The plug in zero. We get to one that if we plug in one, we get 30 which is exactly what again? Since we were going to write everything in terms of teeth, We're going to determine the X in terms of tea and doing so excess two plus t. We take the derivative of that with respect to t we just get DT so d x is equal to DT. L d Y is just a derivative of one minus t with respective team. So the allies just negative DT Okay, now we take our line integral and we plug it back everything back it and everything should be in terms of teeth. So our limit points is from 01 and we plug in our ex. And why are DX dy y everything is in terms of teeth. So if we work this out, we're gonna get four minus T minus four plus 40 plus t squared force cancel and we're left with negative five T minus two squared DT on our integral goes from 0 to 1. So when we take the integral of that, we get negative five t squared, divided by two minus T Q divided by three. And then if we again these limit points we get minus five divided by two minus 1/3 and you'd plug in zero. We're just gonna hit zero. Then again, common denominator. We're gonna multiply. Negative five, divided by two by 3/3 is just one, And then we're gonna multiply negative 1/3 body to over to And if we work out the Alger crap, Rekha, we get negative. 17. You put it by six. So the line integral for the second and going from 00 to 21 is 16 divided by three on the line. Integral bone for 21230 is negative. 17 divided by six. Again, we're going to multiply this by let's say we want a common denominator. So if we multiply this by two divided by two, we're gonna get 32. Was invited by six by 17 divided by sex, which is just five haps five divided by two. And that's our line integral for both segments.