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Evaluate Sc (22y c)dr + (y cy? + 4)dy where C is the boundary of the curve given below:QOMB,Mner15CW55...

Question

Evaluate Sc (22y c)dr + (y cy? + 4)dy where C is the boundary of the curve given below:QOMB,Mner15CW55

Evaluate Sc (22y c)dr + (y cy? + 4)dy where C is the boundary of the curve given below: QOMB, Mner 15CW55



Answers

Evaluate $$ \displaystyle \int_C (y + \sin x) \, dx + (z^2 + \cos y) \, dy + x^3 \, dz $$ where $ C $ is the curve $ \textbf{r}(t) = \langle \sin t, \cos t, \sin 2t \rangle $, $ 0 \leqslant t \leqslant 2\pi $.
[$\textit{Hint:}$ Observe that $ C $ lies on the surface $ z = 2xy $.]

Were given a line in the room over a curve C when we were given the parameters ation for this curve when we asked to evaluate this line integral so the line integral is over the curves c of by plus sine x d x plus z squared plus co sign Y d Y plus x cubed DZ The curve c has Parametric equation are of t equals the vector sign Ti ko 70 signed to t from t equals 0 to 2 pi. So I'll notice that this line integral can also be written as the line integral oversee of f a vector field f where f is the vector field Why plus Synnex I plus z squared plus cosign Why j plus why cubed Sorry not why Cubed execute. Okay. Therefore, it follows that the curl of F is equal to negative to Z I minus three x squared J minus one. Okay, now notice that a trigger than it is Sign of two t is equal to to scientific e co sign of tea. And so, looking at the formula, the vector equation for our curve we see that curve C lies on these surface Z, which is sign of Tim T equals two times X which is sign t times. Why? Which is Kassian t now? Well, let s be the part of this surface that is bounded by sea in the projection of s onto the X Y plane. This is a region D, and this is in fact, a unit disk X squared plus y squared is less than or equal to one. Now, looking at the way our curve this set up a curve C is traversed clockwise when it is viewed from above. Therefore, this induces a downward orientation on our surface s and we're going to use a previous result that simplifies calculating surface integral. So we have that z are surface is a function of X and y, so we could say Z equals g of X y equals two x y, and we have that the function p This will be the first component of the curl of F, which was negative two z which, substituting from our surface see, this is negative four x y. We have that the function que this is theseventies, opponent of our curl which is negative three X squared and our is thief third component of our curl, which is negative one. And also we're going to multiplier integral by negative one for the downward orientation of the surface and therefore by Stokes is the're, um the line Integral oversee of F is equal to the surface integral over s of the curl of f and this is equal to sorry the opposite of the surface integral. And this is equal to the opposite of the double integral over the unit disk D uh, p So negative four x y actually negative p. I should say so. The opposite of negative for X y times the partial derivative of G with respect to X, which is two y minus que which was negative three X squared times the partial derivative Z With respect to why which is two X plus our which was negative. One D a. This simplifies to the opposite of the double integral over D of eight x y squared minus unnecessarily a plus six execute minus one d A and then making switch to polar coordinates. This is the opposite of the integral from 0 to 2 pi integral from zero to the radius of the disk, which is one of and this becomes eight times. And then our cubed times co sign of data times the sine squared of data plus six times are cube times, cosine cube of data minus one. The differential is R D R D theta taking the anti derivative with respect to our in simplifying, we get the opposite of the integral from 0 to 2 pi of right 8/5 Cosine, theta sine squared theta. Plus, this is 6/5 cosine cube of theta minus one half d theta. Get the derivative we get the opposite of and here doing a many U substitution with the fifth times. One third is eight 15th sign cubed of data. And then here cosine cube to take the entire river that this is plus 6/5 times signed data minus one third sign cube data minus one half data from 0 to 2. Pi in evaluating first two terms will drop out, and all we have is the opposite of the opposite of pie, which is simply Hi,

Okay. What we want to dio is we want to evaluate, um, over the curve on the line integral of X y plus, why play Z d s? Um, along the curve R t is equal to to t i plus t j plus to minus two tea. Okay. And t t goes from 0 to 1 inclusive. Okay, so the first thing we need to do is we know that d s is equal to the magnitude of e of T. Um d t. And so first thing we need to do is we know that the of t is equal to the derivative, um, of that curve r t. So this is gonna be equal to two i plus J minus two K. And so, um, the magnitude of e of t is equal to the square root of t squared plus one squared plus a native to squared. And so this is gonna give me, um three. Um, and so Gs is equal to three d t. Okay, so this is gonna be the integral from 0 to 1. Um, X is to t. Why is t and Z is T minus two t and three D t so that seeing a girl we want to do, um, so this is gonna be equal to, um, three times, integral from 0 to 1. Ah, to t squared minus t plus two tt. So this is gonna be three times 2/3 t cute, minus 1/2 t squared, plus to t evaluated, um, 0 to 1. And so when we do that, um, we should get 13 house.

Hello Everyone very are going to solve problem number 36 here. Out off deep tickle toe scientific I kept plus three saying People take up plus Science Square t kick up zero less than a record of less than a record off by by two. If the Kyoto two I plus pigsty plus like kick executor scientists where you could toe 3 70 30 kowtow, same square feet have not They are people do 02 bye bye to science square tape plus sign I was society the plus three saying the kid into because the I plus three costea two plus toe suddenly cause the okay the physical to in the zero to private too. Because the Science Square T plus three costea scientists plus six Science Square t cost it duty which is the Goto Internal zero took bye bye to seven Science Square t cost it plus three by two Signed tau teach they did again for two intervals. Zero took bye bye to seven cause it e minus seven cause Cuban pete plus three by two signed duty DT equal to in the zero to buy back to seven caused the minus seven and do one by four in tow, cause three T plus three quality plus three by two. Signed toe be duty internal 02 Bye bye to they get Sorry. If integrate, we will get life seven Same T minus seven by four into side three T by three plus 3 70 plus three by two Into minus course to be by two 02 by there too. You could like seven in tow one minus zero plus seven by four indoor one by train. Do minus one minus settle plus three into one minus zero plus three by the window minus one day too in tow minus one minus one, which is equal to 46 by 20. 23 by six. Thank you.

Okay, so I'm going to explain the setup of this problem. So I'm not completely sure. Author, house author, design department Who is solvable? I think I'll stop. It's the cinema and explain why First of all that the way to this kind of blind, The whole part of the first things to fill out Yes. So we computed derivative with respect to tee off home Finish off that Megan he appealed by for that group Chairul Thank you. Yes. We'Ll be at this after you simply for using the fact that Science Corp Roscoe's and spirits What? You get this And, uh so that's trying to figure out what was it A crow is We tried to integrate from zero to two parts. Ask you so is heat with a negative three t Oh, Sy Q forty. Why square it'll negative to Teesside Square forty z The s is three squared off too Not funny e t So we have everything set up zero to two pi We put three squared off to you for months and we have each with a nephew seventy coz i cube for teeth Sai square for teeth Hey, so this will be a very difficult. Integral to it. I don't think I'll completed because I have to use our other five different pages, too, to finish it. Let's give you an idea how to how to deal with qualities like this. Well, maybe the things you can do is visit ideas in general when you're dealing with trigonometry, integral high power are usually harder is with the always They're larger and goes, So the idea is you use product to some former to to to write the product of true gumption function into the sun and, as you know it shop the exponential function times trigonometry function without the power is some power in your manage both to Dubai. Use the integration by parts they're being say, still very lengthy process so I can try to show you how to deal with is the first step. For example, co sign square forty size square forty. This will be one over to sigh eighty square. So one over for size square eighty, this will be one over eight minus one over eight. Co sign sixteen t. Yeah, and, uh, but that's not everything we have. We have coast. We have another co sign. Fourth he So we have. So this part's everything I'm writing. Red should be one over eight forty minus one over eight. Co signed sixteen p. Coz I'm forty here. We have to use another product of some form lock in. So coz I off our coast on beta is, uh is co sign off a plus, baby, Huh? Minus coz I off minus pate are over two. We have to divide this. Those are here, I think is plus no anyway. And did you hear? So anyway, we covered it into this and it's off them. We can do the each of them, like, eat like that If eighty co sign Bt this all each of them is the standard power in your congress too. Where you do the introversion by part and do the rearrangement to you'LL be able to, uh, together value off this integral and as you and you have the pocket in here which I think is way too long for the purpose ofthe one video. But I hope I hope this this explains the concepts. And if you want to go on and on, figure out the detail you can You can't do it yourself. Or if you want to just know the approximate value you can use a computer to compute this


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