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

$$\int_{C} x \sin y d s, \quad C \text { is the line segment from }(0,3) \text { to }(4,6)$$

$\frac{20}{9}[\sin (6)-3 \cos (6)-\sin (3)]$

Vector Calculus

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

Okay, So in this question would ask to determine the line integral of X Sign 90 s are given that your curves of the line into rose along the curve. And we're told that our curve is a straight line going from 03246 parents. First of all, we need to find the equation of the parameter very sly. The first thing we have to do is we have to determine the what's the parallel. What direction is this like? Go the parallel direction. That's just simple. Your subtract. So your neighbor, this x one this by one this x to this white and you just threw x two minus x one, which is before what is zero call by, like, two minus why I wanted to six minus three. So we get that the parallel direction. So our line is parallel to four. All right, now we know, uh, our sonnet zero I want spray are a swell right. Why is being now we just plug it into the regions for X and y so packs is excellent plus 80. And why is why one plus 80? Free plug bad. And we get that accents 40 and why is three plus three and now our t sore initial 0.3 So for plug in 0 40 we're going to get X is equal to zero. Why is a 4 to 3 now? If we plug in 1 40 we're gonna get X is equal to four y Z 4 to 6. So really are. T should only go from 0 to 1. All right, now we parameter eyes are mine, okay. And then the next step, what we have to do is we have to write the s in terms of tea, so we know that the arc length is just square root off. D y by D t Square. What's the explain? 90 square TT So do I buy the X If we take the derivative off? Three plus three t uh, with respected T, we take the derivative of three plus 30 with respected two of estrus three. And if we take the girl bit of X which is 40 with respected t that we just get four. All right, so we do this now. The square root of three swear force. So three squared is nine or square to 16. So nine points to 16 is 25. Predict a swear word of that. We have five. So yes is five. Great. So now we can plug everything back, and then we're gonna get on. We're gonna integrate with, respectively. So we plug in our exes so we know that our excess 40 our want is three t bus three at our DSS five DT. If we plug everything back it and we know that our T goes from 0 to 1 Now we get integral. Everything is in terms of our perfect five times four is 20. So we just pulled bread out and then leave. Take the integral from 0 to 1 off T sign three team. All right, so this looks like a product functions of tea and signed three t plus three. So directly we What comes to mind is the integration by parts. All right. And then remember, all our rule is a l i a t e. So this really tells us. Well, you should be so because sign is a trick function sign is a trick function which comes after teas algebra. So a comes before you know, It used to be the algebraic functions are you is gonna be teeth. Okay, so now we know what are you will be so signed. Three to you plus three d t is our Devi. So now you derivative of you with respect to certain used to DT and devious Signed three t plus radiant e And now Wien. We just take the integral that not going to detail how you're taking to get the integral that. But it's just simply your substitution. So where he is just negative with bird close, I I have three keepers 30 technology chain rule says so we're gonna keep our 20 on the outside and then we already know we're gonna use all our, uh, integration by parts for real floor the interior from zero A lot of cheese Sign three Teen. So that's just UV minus Interval VT you. So you ve is just a negative 1 13 co side Three key points three. So we just multiply this by this minus the integral off our feed. You so Phoebe, use just negative with third co sign three t three times. DT. All right, so now now we can integrate Thean integral off. Uh, negative. What third coastline? Three T plus three d t. And again, this will just be you, Substitution. All right, So what we get is the phone. A lot of this right here. It was, but we were waiting to rate what we do. The integration bones. All right. Now, all upper limit is one are lower living zero. So we plonk that in. And then what we get is 20 and then negative. 1/3 coast six plus 1/9 sign six minus 19 Sign three. Again. This is just the flooding. In one event, subtracting have been plugging in. All right, And then if we simplify this, we get 20 times negative. 1/3 coastline six plus 1 96 minus one night. Sign three. And what we can do to simplify this a bit further is we can pull out a 19 So we're going to get on the outside. 20 divided by nine. That we get negative. Three close on six. Plus sign six minus three. All right, so that's our general. That's our line. Integral