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Problem 11 Medium Difficulty

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x e^{y z} d s,} \\ {C \text { is the line segment from }(0,0,0) \text { to }(1,2,3)}\end{array}$$

Answer

$\frac{\sqrt{14}}{12}\left(e^{6}-1\right)$

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Video Transcript

right. So in this video over as to determine the lining to grow up X, he did know anytime he eats the power of Y X or given that, um, along the curve C where C is a line segment from 0002123 First thing we have to do is to write everything in the nineties role in terms of teeth T and D. So the first thing we have to do is for amateurs. This line segment so two parameter rounds it simply where we have to do is x two minus x one calma by two minus y one comma C to minus C one. So, um, it's right over here. We performed this right over here. So one minus zero is one to minus zeros to three line issues. So we get this parallel the doctor, and then now we use our formula for pure amateur ization of excess. Just x one plus 18. Our X one is zero. Most won t or just tea again, Why is why one must be t are why 10 on the on a theist use of zero plus two t which is just to tea and then finally Z and C C one plus C t r Z one is zero. So we have zero plus three to your just three teams and our T goes from zero 21 So we're plugging zero here for tea. We're going to get zero calm under Times Mirror, which is zero and comma three times the euro, which is also zero. So we get a 00 reserve so that we get back our first point and then to get back our second point, we just plug in 1 40 So get 123 which is our second. All right. Now that we determined exploit and Z in terms of tea, we have toe find DS in terms of and then we use our formula for our claim, which is just a square vote off the X by d squared plus Delilah DT Spur was easily by d squared and then we take the square root of that times DT So this thes the explain TT's just wanted The lie by DT is just too and easy by DT is just three. So we have one square roots to spread plus three squared just 14 so the square root of square root of 14 DT So RGs minutes where roots 14. All right, so now we're gonna plug everything back into our line integral. So we have the integral from 0 to 1 t a to the power to tee times three teeth times the square root of 14. So that's just t. Either the pound up to two times 30 is 60 square, and then we're gonna pull the square root for a team to the outside because it's a constant. So finally we get this interviewing right here. This is the square root of 14 times the integral from 01 off T V A to the power of sixties clarity and this one. So we have a T square in the exponents and T, so we can sort of guess that we're gonna have to use the U substitution. So we're gonna sit. Are you people to 60 square? That means the US 12 t t so T v t is you divided by 12? All right, so and then we can also change the limits. So instead of integrating with respect to TV that integrate with respect to you, so are lower, limited zero. We're gonna pluck that sort of six times. Zero spread is just Serie. That's lower than this. And six times one squared is just six. That's our upper living. And then we have you. And then do you divided by 12? What? We can pull that 1/12 to the outside. Now we have a very simple limit. The integral of you know, the power of you, do you? Which is just either the power of you. But our limit points are 60 So the answer is gonna be square root both 14 divided by 12 e to the power of six minus u to the power of zero. But either the power zero is one. So this is our final concert square. It'll 14. Nobody Brightwell eat in the power of six months.