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Question 1[10 + 10 + 10 10 = 40 marks] We can use the equation of curve in polar coordinates t0 compute some areas bounded by such curves. The basic approach is the...

Question

Question 1[10 + 10 + 10 10 = 40 marks] We can use the equation of curve in polar coordinates t0 compute some areas bounded by such curves. The basic approach is the same as with any application Of integration: find an approximation that approaches the true value: While doing some expermenl; class of mathematicians came up with figure with polr _ region R aS shown in the figure below. Find the area of the polr _ region RProve that lim X"y" dxdy =0.Sketch (Using Mathematica) and find t

Question 1 [10 + 10 + 10 10 = 40 marks] We can use the equation of curve in polar coordinates t0 compute some areas bounded by such curves. The basic approach is the same as with any application Of integration: find an approximation that approaches the true value: While doing some expermenl; class of mathematicians came up with figure with polr _ region R aS shown in the figure below. Find the area of the polr _ region R Prove that lim X"y" dxdy =0. Sketch (Using Mathematica) and find the volume the curve formed the cylinder r= cOs y , Zsys3, on the tOp ofthe plane - =-2x, and below by the xy plane: Use spherical coordinates Lind Lhe volume of the solid bounded below the hemisphere p=l,z20 and above the cardioid ol revolution p=l+cos0. Sketch the region using Mathematica:



Answers

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$y=\sin x, \quad y=2 x / \pi, \quad x \geqslant 0$$

For this problem were given the functions y equals four X squared minus one, which is shown here in green and why it was co sign Pi X shown in blue when asked to shave the region in between the two functions, which I did hearing that were then asked to calculate the area of this region. To do this, we calculate the integral from y equals negative. 1/2 y equals 1/2 of the top function, minus the bottom. So this is the integral from negative 1/2 to 1/2 of co sign pi X minus or X squared minus one. Distributing the negative, this becomes the integral from negative 1/2 to 1/2 of co sign pie Axe Mattis four x squared plus one g ETS to find the integral of co sign Pi X. I actually kind of worked backwards. So you know the integral of co sign a sign I wrote Sign Pi X, then divided, divided it by pi because if you do the derivative of sign pi X over pie, you get co sign pie acts, which is the same as the integral of co sign pi X to get sun packs of To get the integral of four x squared. You take the exponents in ad one and then divide by that new expert, so becomes minus 4/3 execute. And then, for the constant one, it just becomes the constant times X So one X or just acts. We then plug in 1/2 her ex. So the sign of pi over two is one. So becomes one over pi minus 16 when you plug one happened here plus 1/2 brax minus parentheses and then plug negative 1/2 in brax. So it's minus parentheses. Negative one over pi +16 minus one hat, one of her pot pie, plus whatever pies to over pie. And then it's 1/2 plus 1/2 which is one minus 1/6 minus 16 So it's one minus 26 which is 46 or 2/3. Therefore, the area between the two curves is to over pi, plus 2/3

The question gives us two curves e to the X and X squared minus one, which I have labeled. Why one and why two, respectively. And the question asks us to find the area between these two curves in this domain. So first of all, I would draw the curves. So here is E to the X, and then here is X squared minus one, and this will be the area we're trying to find between X equals negative one and X equals positive one. So if you do this, I would divide this area into infinitely many infinitely thin vertical rectangles. And the reason why would use vertical rectangles instead of horizontal ones is because all the vertical rectangles would have one side on E to the X and another side on X squared minus one. There will be no cases with a vertical rectangle going something like this, where both sides are bounded by the same curve, and every vertical rectangle here has an equal with or infinitely thin or infinitely small, equal with of D x, and the length of every rectangle is defined by a point on e. T. V X, minus a point on X squared minus one or, in other words, the top curve minus the bottom curve for every rectangle. So using that we can write an integral defined by the top curve or E T. The X minus the bottom curve or X squared minus one. And this equation were expression over here defines the area between these two curves between X equals negative one and X equals positive one. So you solved the question. We just have to simplify this equation until we have a number as an answer. So first of all, we can integrate both of these and that will give us this. And from here, we pluck in the ones and negative ones, and that would produce this expression. And then from here is just simple algebra. So we just simplify all of the exponents and combine the like terms. So after some simplification, we would get this and we could simplify this even more to get this so Eve minus one over E plus 4/3 would be our final answer

For this problem were given four functions y equals X shown here in red. Why will Sanex shown here in blue X equals pi over two from here in green and X equals pi shown here in orange and were asked to find the area in between these curves shaded here in this gray black cloudy color. One of the questions is whether we should integrate with respect to X or respect toe. Why, if integrating with respect to X, you see that it would go from X equals pi over two. So X equals pi with one function constantly over the other with an entire interval meaning that we'd only need one integral to explain the area in this region. However, if we were integrating with respect to why we would have to do multiple in it rolls with one in a girl spanning from life will zero to wyffels one one integral spanning from y equals one toe roughly wipe with 1.5 and another from like was 1.5 toe roughly y equals pot. We have to do separate into girls because along this area different functions are over each other, requiring multiple in a rolls to explain the whole region. Therefore, we integrate with respect to X, so we can only use one inner integral. This interview will be from pi over to two pi of the top function on top, which is X matters the function on the bottom, which is sine X the integral. To get the interval of X, you add one to the exponents into bad by the new exponents, so X becomes 1/2 X squared. The integral of sine ax is negative. Cosa X so negative sine X becomes positive co sonics. We then plug in pie for X, becomes passport over to plus and then the coastline of pious negative one. So passport over to minus one, minus parentheses. And then we put in pi over two for acts so that would become pi squared over eight plus in the coastline of high over to his ears, a plus year distributing the negative and adding them. Together we get the area between the curves to be three pi squared over eight minus one

So, Alva, this cylinder between Well, I mean cylinder Nina. Three dimensional space in our free is described by the equation X squared plus C squared. Is he called the 10 on. Do you want to find the area with the portion portion off the cylinder? That between, uh, why equals to minus one on why equals one? Also, let's sketch how this could look like. So we have here the x axis. See taxes going up? Yes. See, on the y axis going along there, That direction So it increases in that direction. Increase in that direction increases in that. So this, uh, in the XY plane that can be seen as a let's take. Why the plane? Why he called zero corresponds to the explain excessive plane. So order is a circle of radius. Um, Tennessee Goto square root off 10 squared. So it is a circle of radius scores of 10. So you have this very nice circle in the the C X plane. Um, some circle of reduce square it off. 10. That is the length this length order on the all the conditions is that it is this this circle of the cylinder between the goes minus one on one s. So we're gonna have, uh, something standard, probably along y e. So we're gonna have a portion off the ceiling. They're like, uh, BCE, you know? So, uh, well, should look something like this Say something that are goes around there. She want to find the the area of this, uh, off the cylinder. Uh, area. So for that, we can do a final decision. Since citizen two dimensional object in r three, we convert, try, sit in terms off there's off some bible you on some angle. So two pouches So you could say Well, for the angle, we're going to say that exist ical to square it off game that is the radius. Then times cause sign of that angle on. Then why No see, she will be ableto spirit off. 10. Um, sign of that angle sign of, you know, on the novel. Why, it just goes, Why? Why? Why would go to you? Okay. To have these survivals, the angle Thera and you and so well, the the area is gonna be people toe integrate all the bounds for for Sarah are gonna be that, uh, Australia has to go all the way around to make full circle. So the angle Ferre is measured eyes measuring uh, the distance to the X axis. So it will be We have a portion That of things like these. The angle said I would be the single so fed out need to go single from zero up to two pi to have the whole term the whole circle on the unknown, or being go between uh minus one on one minus 11 So should be should be like Holland. So here to buy the area that village the Sierra and then for B B T minus 11 TV on the immunity integrator here the a spark of our little with our view This'll piece here corresponds to the differential of area on dso All this quantity is gonna be well that can be computed as the following determinant. We right, I dedicate e. I mean, how long this role we're gonna right? Alpha that are the relatives of the components with respect to theta I'm along this sec The third role are severely active. All the components with respect to use all the devolved um effects with respectable there would be squared off 10 My sign It's gonna be minus squared off 10 Shine Vera Yeah. Uh, off. So why or zero under a devolved see would be squared off thing cause And then for the for you off this is should be not be better You this by always. You you Yeah, um this this would be York. Yeah, so I mean, I'm just saying it's not It's not really that you So, uh, there to cover the components in respect to you. So they develop X with respect to use you. They had to know why, with respect to you would be one good the Z with respect to you with you So that this vector is gonna be the vector of all these times That so that minus times that zero for Jamie would have Ah, I am No or I It's items. So systems that minus that comes that's a minus spirit of pain, cause now, for J, it would be J that times that minus that times, that's o j zero on the NK. And so these times that minus at times that so minus Okay, I m c squared off 10. Um, I'm sign. Yeah, so that these things vector has alarmed a normal. This victory is gonna be squared off the components squared. Alright, together. So it's critical. Dam Square is gonna be 10 um, scores squared plus and I'm sine squared, which this can be simplified to 10 times. Course the square plus sine square. Very nice. This is people toe one course square person scores one. So these norm the norm is gonna be squared off. 10 Indiana. All the these would be photo. That area area is gonna be equal toe squared off. 10. Um, right. Do you deserve you the sierra? And then you goes from minus one upto one on terra goes from so you occupy. Well, this is ableto so we can pull out the spirit off. 10 internal off. Use you always getting minus one or one. Plans for the development. Just Sarah. Evaluating between to buy on zero so that these will be equal to squared off then thanks. One minus minus one minus minus one times. Um, by minus zero times to fight my minus serious to pilot. So this is equal toe one is minus minus. That becomes plus or you'll be, um, plus once it's gonna be to screw it off 10 times toe by. So it is equal to four squared off 10. And so bye. So that is gonna be theirry a off this question off cylinder. Uh, area. It's four kind of 10. Sorry.


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