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Problem 16 Medium Difficulty

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C}(y+z) d x+(x+z) d y+(x+y) d z} \\ {C \text { consists of line segments from }(0,0,0) \text { to }(1,0,1) \text { and }} \\ {\text { from }(1,0,1) \text { to }(0,1,2)}\end{array}$$


$\int_{C}(y+z) d x+(x+z) d y+(x+y) d z=2$


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Video Transcript

All right, so in this question, were asked to determine that line into throw off. Uh, why, plus cdx prospects was Edie. Why both exports wide, easy. And we're along the curve of C and C consists of two line segments long going from 0002101 And the second line segment going from 101202 All right, so the first thing we want to do, which we want to program a tries his spine. So first, we're gonna, um so see, one is gonna be from 0002101 And to do that, we just do extra minus X one comma. Y tu minus y want commas e tu minus e. So we get that The doctor that's parallel to the line segment is 101 and then not to find the equation the the parameter rised line. We know that X is just excellent. Plus 80 So x 10 on a is one So zero plus one t or just tea. And why is why one plus Bt, which is zero but syrupy, which is just zero a bottle of these. Do you see one plus C t which is zero plus 20 or just and I know T ranges from 0 to 1 because of people again. Zero we get for a T. We're gonna get X zero, Lyon 000 Because our first point again one for T we're going to get X is one. Why is zero and C is one which is our second. All right. So our goal is to write our line integral completely in terms of tea and now we can substitute for X t for y zero fers et But then also another thing we need to write RDX is devising disease in terms of the so DX is just DT, we take the derivative of acts with perspective team, so access to the derivative of TV respected to use just starting team. You're a bit of a steerable perspective. The tea is just zeros of u Y is equal to zero, and then you can see it's just figurative. A busy with respected T nose and see is, uh, derivative of tea with respected T is just one. All right now we're gonna plug this back into her integral forever. Forget everything back door into grow bigger T d t plus two tea time zero plus tgt. Now this is zero. So we just left Teoh t t plus Tedy T, which is just to TVT. And the integral of that is to t squared, divided by two. Now these two guys cancel and we're left with square. The upper limit is one the lower than the zeros. For now, we just subtract. So it's one squared by zero square, which is just so this is our first value. So this is just the line, integral, uh, for our first line segment, which goes from 000 to 10 now for the second light and to grow or line segment goods 1012012 again were First, we're gonna find the vector. That's, uh, parallel to explain segment. So we do extra my sex one common. Why? To minus my one. Come acetyl. It is your zero minus one, which is negative one that, Sir, that's our first really for, uh, X and y tu minus y one is just one minus zero, just one. And now, as the two miners, anyone is just to minus one on one. So this is the factor that's parallel to this life. All right. And then again, just like we did before, we're gonna find the extra wide to see in terms of so X is just x one plus 80. So x one, this one and a is negative one. So we get one plus negative one time steer just one mile it city. Why is why one must be t So I wanted zero and B is once that justice this zero plus one t or simply t NRC is you want for city So see, one is just one and C is one. So this is a B C. C is one we get one plus one tear. Just one city. I don't know. We determined the the X divided t the deer available 112 Stephen respected. He's just negative one DT dear Available tea with respect titties. Just one DT spirit of Zeeland. Respected to use just DT. But again our teams go from Okay, Now we're gonna plug everything back into our mind that was in back and then we get negative one minus two t plus two plus one d t. The one or the negative won't cancel. So we're just asked for the integral negative to t plus two, which is negative two t squared, divided by two plus two. These tourists cancel. And then now we're gonna foregin one. So we get negative one squared plus two, which is one and then we forget. And then we subtract from that we're gonna plug in 0 40 So we get zero squared, negative zero square, which is this bureau plus two times zero, which is zero. So we're just left with negative one plus one since since euro. So it's just one. So now we're gonna add the two line integral values we get. We're gonna add 1 to 1. So one week ago, for the first line segment, the line intervals value is one for the second line segment. The line intervals values also one. So one plus one is two. And this is the value for the line, integral. Uh, along the two seconds

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