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Consider the region R in the first quadrant; bound by the graphs of y = 322 y = 4 - € and € = 0 Set up the integral that calculates the volume of the re...

Question

Consider the region R in the first quadrant; bound by the graphs of y = 322 y = 4 - € and € = 0 Set up the integral that calculates the volume of the region R that is rotated about the Y-axisf" 2 [4 - 3222T 6" =[4 - 322 dr2T 6" =[(4 2)? (322)2] dzNonc of thcsc2T6 [32 4 +x6" [4(322)2] dx

Consider the region R in the first quadrant; bound by the graphs of y = 322 y = 4 - € and € = 0 Set up the integral that calculates the volume of the region R that is rotated about the Y-axis f" 2 [4 - 322 2T 6" =[4 - 322 dr 2T 6" =[(4 2)? (322)2] dz Nonc of thcsc 2T 6 [32 4 +x 6" [4 (322)2] dx



Answers

Find the volume obtained by rotating the region bounded by the curves about the given axis.

$ y = \sin x $ , $ y = \cos x $ , $ 0 \le x \le \frac{\pi}{4}$ ; about $ y = 1 $

The volume obtained when the region in the first quarter is rotated about the bounded by why equal to cube root of X. And that's equal to full time boy access. Way equal to three. So here from the growth you can see that the region abo part of Y axis. As each strip is rotated around the line Y equal to three sweeps out a slice shedded like circular digs with a hole in it. So here you can see that the disk with a hole has a in a radius are sub and equal to three negative curator of X. And outer areas is are out equal to three negative acts upon four. Therefore volume of slice approximate bye time squared off. Also out negative squared off Also in time Delta X. So now here we have to put the value of our sub in and out. So here we get by time it's quite off. Three negative acts upon four. Now you do square off three negative cube root of X. There's the X. Now here adding the volume of all the slice we help and here you can see that delta X. The thickness of each slice tends to zero to obtain definite intrigue. Als since the car what I could do cube root of X and X equals do for why? So here you can see that. Why could too acts upon four intersect, add exit will do 98 zero Positive eight. These are the limits of integration. So here now consider only positive part of X. X is we saw one equal to definite integral of bye time. Three negative Acts upon 4 to the power to now you do three negative cuba ralph X to the whole power to dx from 0 to 8. Similarly region left of the boundary region and what I could do what's up to equal to definite intrigue all of by time three negative Acts upon 4 to the power to negative square off three positive cube root of X. D. X. So volume of solid is We saw one equal to we sub two equal to be. So here week will do definite intrigue all of by time three Negative acts upon four to the whole. Power to negative three negative cube root of X to the whole power to dx from 0 to 8 plus definite intrigue all of three plus acts upon four to the whole. Power to negative X 43 43 plus cube root of X. To the whole power to D X. From -8-0. So it is a final answer.

Bounded by what it will do. Full negative X Squire. Why equal to zero? Ex school do negative two ax equals zero. So here the region is rotated around the X axis so we have to find the volume. So here you can see that graph for this. So here first of all we will slice the region, slice the region perpendicular to the X axis giving circular disk of thickness delta X. And the radius of disk is y equal to four negative X square. So volume of slice approximate by nine Y Squire dry. There's the X. So here we know that radius of the disk is four negative X squares. We put here by equal to four negative X choir equal to buy time. Full negative X square to the power to there's an X. Now we simplify this. So here we get five time X to the power full negative eight times X. Choir plus 16 time there. The eggs now summing over the bones of Kirk volume of solid form will be equal to total volume, approximate sigma. Bye time actually about four. Now you do eight times actually power to plus 16 time delta X. So here as thickness of slides tends to zero. So here we get definite intrigue a lot of by time actually about full negative eight times X squared plus 16 D X from negative 2 to 0. So you can write like that bye time, definite intrigue all of X to the power for negative eight times X squared plus 16 dx negative 2 to 0. So now we have to find the anti davidoff actually about four negative eight times X squared plus 16. So here we get actually power five upon five not to do eight upon three time XQ plus 16 time X from negative two 20 So here are probably zero. So we put here X equal to zero. So here we get bye Time acts to the power. So here practical zero. So I put here zero to the power five upon five. Not to do eight upon three Time zero to the power three plus 16 times zero. *** do bye Time. Lower value is 92. So we put here X equal to negative. So here we get 92 to the power five upon five, negative 8.3 time negative two to the power three plus 16 times negative two. So now we simplify this. So here we get negative pi time. Now you do 32 upon five positive 64 upon three, negative 32. So now we simplify this and here we get 256 upon 15 times. Why? So it is our final answer.

So for this question, what we're finding the volume when um the area between science and coast is rotated about the line y equals one. Um So remember that basically the volume in general is equal to uh the integral from for the given bounds um of the cross sectional area, let's call that a um with respect to X. Um So what we're gonna want to look at is what our cross sectional area would be, and for the record that would be we'd be using the washer method for this. Um because we have two different functions like this and uh neither of them is Y equals zero or anything like that, so um it you don't necessarily have to do this, but it might be helpful to dry out what it would look like. So I'm going to do that. Um basically we're going to dry out uh the graphs for signing coast and Hawaii was one. So let's say this is mhm. We're gonna schedule, how doesn't have to be super neat, but yeah, and let's say this is hi over four. Uh huh. Now, um Basically coast it starts at one and it ends up, sorry, and it ends uh um one over it too. And coast, I mean, sorry, sign that starts at zero and for pie or four, it's gonna end at one over root two. Um And finally We're going to have our line that is rotated about, which is why I equals one and now we're going to look at what what are areas would be, so to do that we're just gonna need our radiant radius for each each like circle basically each disk and uh for for the green the sign. So that's Synnex Mhm. Um That's going to be like this length and uh that would be basically if you look at it well at the top we have one and then we're gonna subtract the Y. Value which is Synnex. Mhm. So that's what one of the ours is going to be like radius radi. I now we're going to look at the other part and that's going to be uh this the radius of like a disc. For for coast and that's going to be uh similar you know except the Y. Value is gonna be co sex obviously is gonna be one minus coast X. That's gonna be the other radius. And um yeah you can see right from this diagram basically since the line is above both of them both the curves and syntax is the one that's like lower, it's going to have the bigger radius so that's going to be the outer ring sort of. So um I'm just gonna you can name them however you choose but I'm gonna call this R. 00. For outer and are I for our inner. Now what we can do basically is um if we think about the cross sectional area then ours is going to be of course I'm from 0 to Pi over four it's going to be and let me just move this down it's going to be our the area of our outer circle. So um of course area of a circle is high R. Squared. So basically is gonna be pi R. O. Squared. And then we're gonna subtract and then we're gonna take our inner circles area which is going to be high R. I. Squared. And then we're gonna have dx and uh I wrote what R. And R. I. R. Is it gonna be um um So we're gonna have one minus sine X. All squared. Then we subtract and we're gonna have pie times one minus uh times one minus co sex square. Okay so there's that now it's just a matter of finding this integral notice. There's probably in each of these as a constant multiple. We can take that out, we can factor that are these and take that outside of the integral. So that gives us yeah high times the integral of everything else without pie good. Um Yeah um and then so I'm just gonna write the rest of that. But in order to find this into grown out where it will be. Well we want to find the indefinitely to go first. So let's do that. Uh we're gonna find this. Mhm. To do that, we're gonna need to expand those. So when you score that first part you get for one minus two sine X. Plus sine squared X. Um and then when you take the other square, the other part. And and to do negative for all of it, you know -1 -1 Uh plus two Coast X. And plus no starting minus coast squared X. And uh these ones will cancel out so and we can rewrite all this. So basically um you can factor that two out of the minus two cynics and the to sign the two Cossacks and that gives us two times coast X minus Synnex. And we have well we have science squared minus co squared. That's you could just factor out a negative from there and you get coast squared X minus sine squared X. Mhm. Does that look familiar? It should because that is in fact coast to X. So we have the integral of this. Yeah uh minus coast to X. And we can pretty easily integrate all of this. We get um two times. Okay integral coaches sign and to grow of negative sign is cokes. So we get that and then uh integral. Of course this sign and make sure divide by the coefficient next when we get minus half of sign two X plus he when we take the definite integral we want me to put the arbitrary constant will get cancelled out anyway. Um so basically what we were looking for we said our V is equal to uh this mhm. So uh well we just found what the indefinite integral is equal to. So basically this is going to be equal to um this entire part. Um Specifically for the bounds from 0 to Pi over four. Okay and uh this this coefficient here to pi we can just keep that factored outside and then we're gonna start and sub we're gonna start we're gonna sub the upper upper bound into all of this part. And as I said before we don't we need that arbitrary constant for this. But yeah and then we'll subtract and something lower bound in. So what we get is um two pi times um ah oh sorry actually that that too can't just be factored out because there's no two isn't here. So yeah, we're just gonna keep that factored of pie outside of it. But yeah we're gonna have high and then we're gonna have two times sine of the upper bound is planned before. So sign of five or 4 plus coasts of five or 4. Mhm -1 of the sign of It's pirate four times 2 pi over two. And there's that now we're gonna subtract. And we're gonna so the lower bound so we're gonna have to Times The sine of zero plus coast of zero mm -1 of the signed of two times 0 is zero. Okay you can start by simplifying things, okay sign of zero is zero. So we can get rid of that and this is just one. So really at this part it's just too and then sign a zero is zero. So this whole part is zero. Um Now uh for this part Sign of power for and coast of power for those are both 1/2. And also sine of pi over two. That's just one. So because that's one you know what we need that um basically what we're left with is this Uh we have one over root two Plus one Over Root two. Mhm. And then uh minus half And then -2. Okay when you add those two fractions there we get We get to over a two. and that too over root two. That's just rude to and then minus half minus two, that's minus five or two. So we have our answer but just to be clear our I'm gonna rewrite it. Our final answer is V. Is equal to high times two minus no sorry to route to -5/2.

Eso looking at these functions. Why equals two n y equals four minus x squared over four. I could Graf about enough that would really help anybody. What you want to do is find the points of intersection, and then we're gonna revolve it then, eh? So what I would do is as subtract two to the right side so I could add X squared over four to the left, and then you multiply so you got X squared equals eight. Cross multiply, if you will. So X is equal to plus or minus the square root of eight. Eso those air your bounds in this problem. So a negative square root of eight. Some people might write those two or two, by the way, but I'm gonna wait to the end to mess with that on. Been positive. Um, so if you need the visual because it is helpful, Michael's two is here. Thea other function is a parabola appear so you can find the point of intersection. Reason why I like showing that that problem is we're revolving around this axis the X axis. So there's definitely an open space, a washer, if you will to do the washer method because we only want this area being revolved. So what I'm talking about is the washer method, because there's a hole in the middle which tells me to do this process. And so what you want to do is do the upper function. Um, you know, I guess what I'm saying is, you want to find the volume If the upper function went all the way down to the X axis, that's that four minus X squared over four. Sure. Uh, minus Taking away the volume of mail. Do this in green. This rectangular box that's being revolved. And you can imagine that rectangle going around and taking out all that space. Well, that function is just to so as we simplify this, what I would do since you're paying for this is I would simplify that for that foil. Basically so 16 and then four times 1/4 we could be one, and we have two of those so two x squared and then it becomes plus X to the fourth over 16 and then minus four. Well, dx, I've already written a lot. So let Z instead of writing 16 minus four, let's just write down 12 right there. So a better math teacher would say to make sure you write it all out. So now let's do the anti derivative, which is 12 x Add on to your exponents divide by your new exponents. Oh, jeez, I don't Even when you divide by five, it's just going to go into the denominator here. And 16 times five is 80. I'm doing math correctly from Negative Root eight to positive Route eight. Oh, my goodness. This is ugly. So now you get to plug all of that in there. As I'm looking at, I'm gonna trust my answer key. Um, that they simplify and they do a lot of other math that they get 448 e trust my calculator route to pie all over 15. So double check type that in, make sure you get the same answer


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