5

Xsen W T dr 2 € 70 2 sen a 3r X + 48(a > 09 4sen ax dx 1 2 Te Ed cos a ~+4(a > 0...

Question

Xsen W T dr 2 € 70 2 sen a 3r X + 48(a > 09 4sen ax dx 1 2 Te Ed cos a ~+4(a > 0

Xsen W T dr 2 € 70 2 sen a 3r X + 4 8 (a > 0 9 4sen ax dx 1 2 Te Ed cos a ~+4 (a > 0



Answers

If $(a, 0) ; a>0$ is the point where the curve $y=\sin 2 x-\sqrt{3} \sin x$ cuts the $X$ -axis first, $A$ is the area bounded by this part of the curve, the origin and the positive $X$ -axis, then (a) $4 A+8 \cos a=7$ (b) $4 A+8 \sin a=7$ (c) $4 A-8 \sin a=7$ (d) $4 A-8 \cos a=7$

Making us all this don't negotiate post we just integrate it back to us with this constant then we in the respective taylor soldiers coming out to be leader papaya that'll be asked him over to from little girl 81 blood boston. What does that have to be 1/2. It's going to be uh we'll have your smart to be a script one plus 40 them. All good just be a war there now this is this is back to data now I'm going to just integrate this so how we can integrate this mouth next work on this will be carried out is equal to is spread over two integration from there to find Which manages one plus to go to class to both either did it which we have uh your head now so was coming out to be Is still over two there but by one plus not hospital he has given us one plus plus who's the leader or will be that is plus Canada plus people theater they did on the other back to listen to it is now you got a spare over to this will be one plus 1/2 3/2 to 3/2 peter plus we have trying to get our all that just to find hey that's the after all you have to buy but he doesn't Work on this we get is still over two then 3/2 3 pi over two Plus trying to find that gives zero who signed by that years later -0 0 and negative there so we have to be three by or four. Yes. Well, let's take it down. We buy our four is good support 25. Well, with option being anger.

One day we're going to start pretty indecorous zero but after the are equal integral zero but X squared, right, Plus ex way did in Duke minus toe science t I plus Tau Costea into the equal toe We get, like, 02 point minus to take the square scientific this bellick flee cause the the here alone are equal toe cause being I plus toe side the that's it Execute toe to cost the by equal to toe saying he integral zero toe fight minus 24 cause quality Thank t plus doing the four costea scientists Costea the Intel zero toe by minus eight cause quality 70 plus eight, of course quality sci fi duty equal to zero. Thank you.

In this problem of line integral. We have to evaluate F. Daughter there along the patsy. And now we have given the function F. Of X. Y. And Z. Is equal to X. Is square. I. Plus why is square J. Plus 30 square K. And The courtesy. It's such that artie is equals two two scientists I plus the cost TJ to cause T. J. Bliss one divided with two which is half the square K. And the value of T. is wearing from 0 to Pi. Now first we have to evaluate dear. That means we have to differentiate itself differences of sinus costs. So this is to cost I. The transition of course is minus sign. So we have to Ready this -2 aside. Did you? And differences in of T. Square is to T. So this is where they would be half multiplied with two T. Z equals two T. So plus tiki. Now FDR data from here we say that X. Is equal to to 70. Why is equal to two costea. And that is equal to one divide with two T square half of X. Wiser. So F. Of X, Y. And Z is equal to X square. That means to scientific Holy Square. Which is equal to four science quality. Why is square? So this is here to coast the holy square. So which is four Courses Square. T. and 30 square. So this way you would be That is equal to one divide with two T square. So there's one divide with food T. To the powerful and this is I. And they said G. And this is key Now F. Daughter dear that means here also term but we didn? T that in four. Sine squared is multiple. Early to cost is so they said equals to four. Sign is square. T. Multiplied by two. So this way they will be four multiplied by two which is eight. So eight Sinus Square T. And cost -2 signed a multiple with four. So they said -8 course is square T. scientists and one divide with 42. The powerful is multiple. Pretty so this is What they were with 42. The power five and DT now we have the integration limit we can replace it with zero to pi. So this is zero to pi. Now we have to integrate it. So integration of eight. Sinus square T. Cost differentiation of Sinus cost. So we can write it as it divided with three signed QB. And yeah And differentiation of course is minus sign. So this way they would be -8 divide with three cause Q. T. So this will be minus minus plus and one divide with four. So this value would be When they were with four T. to the power six divide with six. Now limits limits are from 0 to bye. And now when we put upper limit so sign pi is equal to zero and caused by is equals two minus one. So this valuable b minus one cube. That means minus eight divide with three. Now this is t putting the value pie so this value will reply to the power six. This age Try to the power six divided with 24 and minus lower limit, putting the value 0700 and cost zero is equal to one. So this age It divided with three and putting tichkule 200. Now when we evaluated so minus eight w three minus 80 value three. So this really would be pi to the power 60 were with 24 minus 16 divided with three, which is the right answer. Yeah.

In this problem, you're given the shown information. And our first step in this might be to make sure that our doctor field F is conservatives. And the way that we want to do that, it's set. This here equals some function p and this component here equal to some function. Q. And we're gonna make sure our shoes is equal to the pressure that the partial API with respect to Why is going to be X times two. Why? And is that equal to the partial? Que with respect to X? Let's see, we get two x times. Why, sure enough, those are equal. So it is conservative. So now we confined a function f where the vector field capital after is equal to the grating vector of lower case in the way that we're going to do this is to write our components E no. So when we sold it, you get explain while I squared first. So, um, function wine because the derivative of the function Why, when we take a partial of this in terms of X, would be zero, and now we're going to take the partial of this new function in terms of why And when we do that, we get 1/2 x squared times two y plus G crime of why Now the two and 1/2 are going to cancel out. We're gonna go X squared y plus g promise wine and we're going to set that equal to our function Queue up here. And so we get equals X squared. Why? And from that we can see the G prime of why is equal to zero. Therefore, when we right zero de y in a function or an incident girl Excuse me. We know that G of wine is equal to some constant K. Now, I'm gonna put that into the G of riot there and see, uh, 1/2 X squared y squared plus K is equal to the function f of x y that we're looking for. And so that is our answer to part A. Now in part B. We're going to evaluate this interval along R A. T. Which is this. Now The way that we're going to do this is that we're going to that this component here equal to X and this component here equal toe wine. We're gonna put in one and zero and are just solve for X and y for our endpoint in our beginning point for when we saw. And so, if you do that you see that have tea plus sign of 1/2 fizi equal to x. And we have the tea because coastline of 1/2 party is equal to why and our T goes from 1 to 0. So first I'm going to plug in one. And when I do that, I see the eggs is equal to one the sign of 1/2 pie or just pi over two. Sign of pi Over two is 11 plus one is two. So again this is T is equal to one and then why is going to be equal to one? So why is he going to one plus co sign of pi over two Costa on If I were to zero So one plus zero is one. And so our first point to plug in to our potion effort. Some X Y is going to be 21 and we're going on a cell for the second point now, which is going to be when t is equal to zero. And when she is equal to zero X is going to be equal to zero, but a sign of zero, which is zero. And why is going to be equal to zero close co sign of zero, which is one as our second point is going to be 01 now because we're going from 0 to 1. We're gonna take our point when she who's one and get the results when we pulled into the function. And then we're going to subtract the result from Army Talking T equals zero in the point to get them to 01 And so what have expired? Why squared plus K. Now I'm going to not include the constant, because when you out of constant and subtract that Sam Constant, you're always going to get zeros. You never have to include the constant unflagging in points. And so we get 1/2 x y squared. Anyone has X squared. Why, I swear and you're going to want to that we see that we get one house times two squared times one square to square. This 41 squared is 14 times one is one times 1/2 is to. And then when we plug in zero and one, we're just going to get zero. And so we're going to get to minus zero, and that is equal to two. And that is our answer to hurt. Could be. And again we got this from plugging in she and two r T to get our exes and wise and get those coordinate pairs. And I'm fucking those into the function from for a and subtracting the beginning point from the end point to get our final answer.


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