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Find the volume of the solid generated by revolving the region bounded by y = 2Vsin X, y = O,and X1 X-axisand Xzabout theThe volume of the solid generated by revolv...

Question

Find the volume of the solid generated by revolving the region bounded by y = 2Vsin X, y = O,and X1 X-axisand Xzabout theThe volume of the solid generated by revolving the region bounded by y = 2Vsin X , y = O,and X1 cubic units (Round to the nearest hundredth:)and X2about the X-axis

Find the volume of the solid generated by revolving the region bounded by y = 2Vsin X, y = O,and X1 X-axis and Xz about the The volume of the solid generated by revolving the region bounded by y = 2Vsin X , y = O,and X1 cubic units (Round to the nearest hundredth:) and X2 about the X-axis



Answers

Find the volume of the solid generated by revolving the region bounded by $y=\frac{2}{\sqrt{3 x+1}}, x=0, x=3.5,$ and $y=0$ about the $x$-axis.

If you need a visual for this problem, X squared plus one looks something like this. Um, and then the other function is negative. X squared. Plus two x plus five. So, um, guess as I'm looking at that, it's gonna go up something and come down. And, uh, I don't feel like just trusting right now because they give us that X equals zero and X equals three that the points of intersection. I'm gonna be outside. Otherwise we might have ah, slight issue and solving this. Um, So again, I'm gonna just trust those bounds, but we're revolving around the axis X axis right here. So if we look at the area that we're revolving, there's definitely ah white space between the access we're evolving around and what we want. So that tells me that we have this washer method of some function squared minus another function being squared and over the X axis in terms of DX. So I like to look at the bounds that they give us zero on three and then you want the upper function that negative x squared plus two X plus five. She gave myself more room and then the other function is X squared plus one. And if you're a teacher, lets you go to calculate that should make this problem really easy. Um, but I don't know if if you would be required to do this, because you could. It's not foiling now, but X to the fourth. Let's see the next term. Yeah, this is going to get busy. I guess I should just show the work because remember, multiplied by yourself is what it means to be squared. So, like, here I would get Ah, negative four x cubed. But then so I negative to execute and then another one. So it's negative. Four x cubed on denied. Have a negative five. See, this would be, ah, positive four. So I think of one and then another native six x squared. And then I'd have a 10 ex with another 10 x will be 20 x and then plus 25. I hope there's no typos there. Now this one would be X to the fourth. And then there'd be two x squared and then plus one. I need to subtract each of those. So this actually goes completely away. So we're looking at pi. Sorry, my work is messy. I didn't realize how difficult this is going to be. So add one to your exponents divide by your new exponents. You got to go to a calculator anyway, so I don't know why. I just didn't do that to begin with. Uh huh. From 0 to 3. Don't forget about multiplied by pi. Now it's nice. Is this zero? Subtracting a bunch of zeros is not going to affect your answer. So I'm just going to my calculator. Make sure I'm doing this right. They have three to the fourth, minus eight thirds times three to the third puts 10 times three squared. Just 24 times three. I got an answer of exactly nine. Hi. Which does not match the answer key. So I had a feeling this would happen. It's a quick fix. Yeah. I mean, it's sort of a quick fix. I should not have assumed that three was gonna be in here. What you find out is that X equals two is actually the bound. So what you have to do is all that same work and actually just do it from 0 to 2. All right, So as and the reason why you do it from 0 to 2 is because that will find us this area. And then what you need to dio is add to it and negate all of those values. Because then the other function is above really hope. I had resumed and I did, um, so that flips everything and then do that from 2 to 3. And yeah, as you add this piece together with this piece, I'm gonna jump to the answer now. 277 pi over three, please. Double check that arithmetic is correct. 277 pi over three.

Okay, So the trick to this problem is recognizing that, uh, your bounds of Mexico zero in Mexico. Six. If you were to plug them into your function two over X plus one. Yeah, that no matter what you plug in for X, you're going to get a positive number. So if you're looking at the graph from zero Yeah, zero to a vertical line of six, you're going to get a positive. No, I don't know exactly what it looks like. It's gonna be a positive. And they also give you the other bound of like, 00 Which is also important because if it's Weichel zero that were rotating around the X axis, extend this, give you a little rotation symbol, Um, that it's the disk method, because the area under the curve is touching everywhere. So pi times that function squared. And we're talking about going from 0 to 6, so that's just setting up the problem. Now let's rewrite this. I think that's helpful is you have four over that quantity of X plus one being squared. Reason why I think that's somewhat helpful is because you can think about Ah, I mean, I could show you substitution, but it's pretty straightforward Is this quantity is to the Negev second power. If you put into the exponents, maybe that's what it should have done. But anyway, you to square both pieces as well as trying to show you. So the anti derivative Oh, refuse myself by saying that because you have tow add one to this exponents of negative too. Because if you put it up here, we'll just rewrite it X plus one to the negative second power. So that's no longer here. Um, so add one. But then you have to divide by your new exponents. Don't forget about that pie in front, and we have to go from 0 to 6. Um, and this derivative matches this because if I asked you for the derivative, if you bring the negative one in front four times, negative one is positive for, and then it's become to the Negev second power, so it sticks into the denominator square and then the derivative of the inside. It's just one. So what I'm saying is now you can just plug in negative for over six plus one to the first power and a minus, um, becomes negative force of plus for over X plus one. So zero plus one. Excuse me to the first power. Yeah. Um, Now, if I were doing this, I would get these to be the same denominator. So 4/1 is equal to 28/7 and adding that together, we give you 24/7. And don't forget about that pie, which is part of the disk method, which is your correct answer.

Eso There's two things to be careful with on this problem. We're revolving around the Y axis. And if I were doing this problem, first of all, I would rewrite it by distributing that three in there three times. He was six minus three X. So I would graph that first of the Y intercept of six. Yeah, on down 3/1, not 3/1, then down 3/1. Um, so we're looking at this, and they also give you the bounds of X equals zero. And why call zero? So we're stuck in the first quadrant there. Now, The other thing Thio remembers, we're revolving around the y axis, so that needs to be in terms of why? So you can see your y values are going from 0 to 6. Um, and then the other piece I would mention is you have the disk method? Yeah. Um, because the area is touching the access your rotating around, so just pi times radius squared. And then the other thing to remember is you want to solve for X, So why minus six equals negative three x and divide everything by three there and you're looking at an equation of negative one third. Why? Plus two. So if you want to know how I would do this problem, I would foil that out and get 1/9. Why squared? And I'll be minus two thirds blind, minus two thirds y. So it's minus four thirds. Why? And then plus four. And this is easier for me to do, because then, as I'm looking at that, well, you add 12 Why s that's why cubed, um, divided by three. That will stick it into the denominator. Except I'm doing bad math right now. Nine times three is 27. Um, same thing as you add one to your expo. But when you divide by 24 divided by tourists two of two thirds plus for why? And you're going from 0 to 6. What I like about this problem is this zero means nothing to me now. I would also do as you plug in six and may change 27 to be three times three times three. You can see that you have you know, too times, two times two. So you're gonna end up with eight and then to Sorry. Six square to be 36 on divided by three is 12. Um, Times two B minus 24. Uh, then four times six is 24. Maybe I didn't do this. Judge justice about this zero is if I plugged in zero and for all these could be zero, um zero minus zero plus 00 So that's why I mean zero means nothing to me. Now, as I'm looking at this, this minus 24 plus 24 or cancel. So you just lift with eight pi is your final answer. Double checking that. That's correct. We are good. Hey, pie.

Okay. What we want to do um is we actually want to find the volume of the region that is founded by y squared is equal to two minus x squared over one plus X squared on the interval from 0 to 1. Um And so the first thing we need to do is any time we're talking about a bounded region is to go ahead and just do a quick sketch of kind of what that is going to look like. Um With this y squared, we know we have a part of the graph in the first quadrant and a part of the graph in the second quadrant, so if x zero, y squared is four um and so if I think about that's going to be up here at plus and minus two, Um if X is one now we have 1/4 Y squared is equal to 1/4 Which means that um we have plus and -1 half. So our region is going to kind of look something like this and of course is curved and we're evolving it about the X axis. So remember we are doing um um volume is equal to pi r squared. Um and if you notice that when I revolve this region, it's almost the same as just revolving this rectangle Or representative rectangles that go from the top of this to the bottom of this. Um and so this is gonna be um volume is equal to pi and we're going from 0 to 1 of our squared which is that top function which is going to be just the plus of these two minus X over one plus x. And of course we're gonna square that so that still becomes a tu minus X squared over a one plus x squared T X. Um Now now we've got to figure out how are we going to integrate this? Right? And so the first thing we need to do is probably multiply out that numerator. And so let's go ahead and do that. So we have a 4 -1 X. Plus and X squared over a one plus X squared dx. Um Break that integral into parts. So we're gonna have and for and let's go ahead and, yep Mhm let's go ahead. Let's bring that pie all the way out. And this will be four times the integral from 0 to 1 of one over one plus x. and where square that DX minus the four integral from 0 to 1 of X over one plus X. Squared. And then plus this in a girl from Erdogan have X squared over one plus X. Squared. Okay. And we noticed that we're gonna have to do multiple multiple things here. Um This I'm just gonna do a use up So we're just gonna for this first integral we're just gonna let You equal one plus x. And therefore D. U. Is equal to D. X. And so this will actually change to four times the integral. And then let's go ahead. That's gonna be for 1 to 2 of you to the negative two teams which we know how to integrate. Okay now on this one now is what we're probably going to do is to go ahead and do partial fractions. So on this one we're gonna do partial fractions. And so when we do the partial fractions um we're gonna end up with minus. Um we're going to end up with minus. Um the integral from 0-1 of four over one plus X. D. X. And then plus the integral from zero Erdogan of 4/1 plus X. Squared. And then on this one we're actually going to do a use of. So in this one we're actually going to do you so we're gonna let u equal one plus X. D. U. is equal to DX. But then we're just going to do X. is equal to a U -1. And so when we do that we're going to actually end up with some unique things. And so this one so on this one this is going to be yeah plus in here we're going to have the integral from 0 to 1 of U minus one squared up. And we're going to go from 1-2. Here is 2122 and this is going to be a you square to you. And then of course we are going to multiply so we can multiply this out and then reduce our fractions down. Um And when we do that we actually end up with some unique things. So this will be hi um And this is going to be negative four One over you and we're gonna evaluate from 1 to 2 then this will be minus four natural log of one plus x. And we evaluate that from 0 to 1. Then this is very similar to what we did here. Um and so this will be -4 one over you From 1- two. And then on this one when I multiply this out and reduce down we actually get um we actually probably get a I think it's gonna be kind of like a plus A. I don't think I have a four on there do I am. So this is going to be a You evaluate from 1 to 2 -2 Natural log of you for 1: 2 -1 over you. From 1- two. And when I do all of those substitution we actually get um a pie And then we're gonna end up here with uh 11 over to because here will end up with a kind of like a one plus a one half which gives me a three halves. And then of course I have these over here as well. So that will be 11 over to And then we have a -6 because these are the same. So six L. N uh two. And so there is my volume.


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