5

Find the volume of the solid generated by revolving the rgion bounded by the curve y Un * the X-axis; the vertical line x about the x-axis ale - nle2 _ 1) C) Zale2 ...

Question

Find the volume of the solid generated by revolving the rgion bounded by the curve y Un * the X-axis; the vertical line x about the x-axis ale - nle2 _ 1) C) Zale2 _ 1) D) 7e

Find the volume of the solid generated by revolving the rgion bounded by the curve y Un * the X-axis; the vertical line x about the x-axis ale - nle2 _ 1) C) Zale2 _ 1) D) 7e



Answers

Find the volume of the solid generated by revolving the region
bounded by $y=2 x-x^{2}$ and $y=x$ about
(a) the $y$ -axis,
(b) the line $x=1$

Yeah. The property's drone is so this is it on. This girl is off about my sex. So we have a plan of the area. This is No, This is one. This is June. So But the problem? Yeah, using the So we have by bigger love never explain about my text. Yes, this is given us Dubai? Yes. Beautiful sex minus one minus one. Do the Bartman ethics or what? Minus one days now for the solving this integration. We have this. You buy 96 96. Last minute six over management. This is equal minus X beauty parlor tricks. Six. Goodbye record. We have minutes went by my next one minus the video when it's wet. Minus go, Linus minus Dubai given us bye. Minus good bye. Minus five plus two people toe minus. Find he No, this isn't giving us for what he did. One minute. What money? That this even as good by you might have to do with it. This is that part of the government beyond the problem, it sticks. They by General one limit one. Wanna sex into beautiful aromatics, dears? Oh, just a form would saw one man in a car minus six beers. This is given us one my mistakes into fine effect. What? Minus one minus? There's no Be a part of the job. One minute. Six. You do it for my sex. What? Minus one deals. No, this is funding. You get your house, My self. One of my sex do through the bar. Line of sex. Bless you. Minus one. Uniformed men and sex This is your gun off. Minus one plus x into my sex minus Good on minus six over minus one. Sister Gina's X minus one interview about my no sex. Bless your bottom in Essex. What did you do? Enough. We're gonna fix you one plus X minus one. This cancer's it become. It's good for my sex limited. You do anyone? No, we have the morning. It was goodbye. That's your bottom In sex, you don't know. The limit is the You buy one. Get one minus Well minus zero is equal. Goodbye. Did you want to buy? What? Me? This is the inside of a part and this is over already. When the drums

So today we're going to be using integration. Solve questions on solids of revolution. So really problem on the volume of the solid generated by revolving the region pounded by the proble y equals X squared and the line y equals one will first start with the line y equals one is the axis of revolution. So the first step is to draw the curve. I'm gonna start by drawing y equals X squared. I like Teoh Just all the points on the plot. So go 0011 24 Then the same for the negatives. Negative 11 negative. 24 See how good I can draw this curve Now you and not too bad there, then Michael's one. That one's a little bit easier for me. All right, so we got Michael's one here and in blue. We have y equals X squared. Okay, so the region bounded by That's gonna be in red, which is this region here. So the axis of revolution here is going to define which method we used to do. The integration to the fact that Y equals one is one of the bounds of these curves that's actually going to be that same plot and green. So since the axis of revolution is one of the bounding curves, that means that we're going to be using the disc method already. So after we've drawn, the curves were determined The radius direction. Here the radius direction is always going to be the opposite or perpendicular to the axis of rotation. So your access of rotations here it's parallel to the X axis. So we're going to have radi I and the extraction. Okay, so when you're doing the dis math that you got to think of this bounded curve as an infinite some of all of these very, very small disc sense what the disc method is. So if you think of this curve here that we have, imagine that's the same one. We have all these very small, infinite discs that were going to some across the entire curve. So, as you can see, this radius is going to change. It's going to be a function of X, so you want to determine what that radius is in order to solve your volume function again for the disk method, while you it's going to be the integral oven area. Some Dover height here. We're gonna find the area of each of these little discs as we go from left to right. And your height of each disc is going to be that differential X. This is your differential here. So them when you're finding the radius, that's going to just be the bounds of the two curves. So as you go from left to right, you have y equals one, and then you have a curve X squared, so your radius always going to be the top curve minus bottom curve. So we're gonna have R of X because the radius says we said earlier, is a function of Well, your position on the X axis is gonna be the top curve, which is why equals one subtracted from the bottom curve, which is X squared, us going back. Teoh geometry area of a circle, Mr Find is hi or squared. We'll hear now are our radius isn't constant anymore. It's ah, it's a function of X. We're gonna have to square this function here, So r of X squared is going to be one minus x squared squared. That's going to equal one minus to x squared plus X to the fourth. It's really back it out to give me a little bit more space now. So to solve this, uh, disc method integration here, we're gonna say the volume of this solid revolved about the curve y equals one is equal toothy some of all of these areas across this curve. So it's gonna be an integral to define your limits of integration. You're gonna find the two intersection points on this curve, and this one's a little bit simple or some might be a little bit more complex and require some actual algebra. But the two intersections or the points 11 and then negative 11 Since we're integrating in the X direction, we're going to use the X coordinate, that's gonna be the integral from negative one toe one of your area function in your areas that pi r squared. Going to say pine on R squared is going to be one minus to X squared plus X to the fourth. Let's make sure that four looks a little bit better, and then area times your height, which is your differential eggs. Since, uh, pie is a constant, we're gonna pull that out same time is performing or integration something to say that that's pie. We're gonna say That's X minus 2/3 X cubed plus 1/5 X to the fifth, evaluated from one to negative one. So when you go ahead and apply your confidence of integration, your pie times could be one minus 2/3 plus 1/5. Let's make sure that fifth looks good subtracted friend. Now we're putting the negative one since one plus 2/3 minus 1/5. Simplifying here pie, Uh, minus minus is plus there. So that's two minus 4/3. Do you think this getting a common denominator here that will get us to pie Times I always come to dominate. Here is 15. I have 30 minus 20 plus six all over 15. Simplifying here. Let's give us 10. 16/15 so that 16 hi over 15 volume of that curve is going to be a 16 pie over 15 sets curve about the line like was one. Let's go to the next one for us. Gone ahead and drawn the curves for us this time just to kind of speed up this process and our new line of revolution is y equals two So since your axis isn't actually bounded by that curve, you're gonna have this space here that is empty. So if you can think about this, your new curve is gonna look something like that, or this is also part of its He have this inner radius. It doesn't have any solid, so that can't go to the volume. So this man tha that we're gonna look at is the washer method. Any time that you have a void in your, uh, solid of revolution, you're going to use the washer instead of the disc method. So very similar formula. You know the volume it's going to be pine r squared my the smaller inner radius times I hear it's gonna be that differential again. So your two curves here bounded by big or which is your bigger radius, which is from your access of revolution to your bottom curve. Actual Trauger here in black, and then your inner radius is going to be the smaller curve here. That inner radius is actually going to be a constant because it's going from the curve Y equals two twi calls one. So let's go ahead and get these two curves evaluated so big are I want to be a function of X that's going to be top curve minus bottom curve. So that's why equals two minus curve X squared. So that's two extract from X squared man. Little are it's actually not a function of X. It's just function. It's constant. So we're gonna say R equals. Why equals two minus one, which he calls one. So let's go ahead and square these two. The inner radius is a little bit simpler. R squared is just gonna be one square, which is one R squared of X. It's going to equal for minus four X squared plus X to the fourth Again, these air, the infinite sums as you go from your same constant of integration here, that's gonna be negative. 1 to 1 again. That was found in the previous problem. A. By looking at the intersection of two curves like als X squared y equals one. I swear to find all of these infinite, uh, infinitely thin washers. You're going to some of the areas of each of these to establish the volume. That's the same. Was looking at washer here. We're gonna rotate that access that would look like? Yes, where your radius on the inside here is going to be a constant one in your larger radius. It's going to be a function of X established by the curve two months X squared. Let's go ahead and work through this problem. So the volume it's going to be the in a girl from negative one toe, one your area times by cherry There by your height, so R squared minus little R squared is going to be for minus one is three minus for X squared plus X to the fourth the eggs Applying the end girl here. We're gonna say that that IHS bring out that pie Constant three x asked 4/3 x cubed plus 1/5 X to the fifth. It's pretty sorry. Five and then your costs integration gonna be applied over here. One a negative one. So now going ahead and applying these ball He was going to be pie. Applying one is going to be three minus 4/3 plus 1/5 you subtract by the other B minus three get a one here is going to be 4/3 B minus 1/5. Gonna go ahead and apply the signs Simple. Fine. We have I have six minus third's plus 2/5. Simplifying Begin. Please come. Denominators could be 15 3.5, My times six times. 15 90 my ass. 40 plus six all over 15 which simplifies down to 56. We were 15. So your volume of this bounded curve we're both about the line y equals one. This 56 by over 15. All right, last one. Now we're gonna go about the curve. Michael's one. We're sorry. They curves y equals X squared and y equals one. The bounded region between these two revolt about like was negative one. I've gone ahead and drawn. You're bound curves as well as the axis of rotation here. So if you look at it this way, gonna have another curve bounded over here. Go ahead and draw this in for you for the curve on the other side of the access of revolution, you have this well hourglass shape here, a void in the middle. It's a little bit different from the last born, uh, same method again because your access of revolution is not one of the curves bounded by the solid revolution. We're going to use the washer method again. All right, so same formula that we've used all along here volume is going to be the area times. By your height since its differentials, we're gonna have the end of an intestinal e small. I be your differential on X s. So let's go ahead and find your to, uh, radius curves. So big are it's going to be a constant here because now you're talking to your axes. Revolution is actually constant. That's your big are a little r. I know. I've switched up the colors on us. Now it's going to be that inner radius here. Your inner radius is actually the function of X on this plot. So let's go ahead and get big. Are since its constant here, you constant radius. It's going to be to we're sorry not to. You're curve is going to be one finest negative one, which he calls too. Your inner radius. Little are going to be a function, uh, ex. That's gonna be top curve minus bottom curve. So that's going to be X squared. Minus negative one So minus negative one going to be X squared, plus one. It's going ahead and doing our squares here. R squared of X. It's going to be X to the fourth plus two x squared, plus one big R squared going to be two squared just for supplying our volume formula for the wash method. The is the integral from negative one toe. One constants haven't changed throughout this problem because it's been the same bounded region. So you're bounded. Region is going to be the intersection of the two curves, which is negative 1 to 1 times by your area, which is going to be I times big are squared, which is for minus little r squared. That's going to be minus X to the fourth Linus two X squared minus one time Sure, Infinitely small high d X Trying to simplify here for me we have by That's gonna be three in this two x squared. I'm not sex to fourth DX. Go ahead and apply my integral. It's going to equal pie times three X Here we go. Yes, 2/3 X cubed minus 1/5 x to the fifth applied at one negative one. Back it up a little bit So that's going to be by I'm spy three. That's 2/3 my ass. 1/5 subtracted from minus three. You have a negative. Negative is a positive. So that's plus 2/3 negative negatives. Another positive. So it's plus 1/5 simplifying. We have pie. Go ahead and apply the negatives here. That's plus plus minus minus. Really positive. Six negative. 2/3. Minus 2/3. Minus 4/3. Minus 1/5 plus negative. 1/5 is minus 2/5. Simplifying here. This is similar to the previous one is going to be. But I was coming down. Mayors 15 again. Yeah, 90 minus 40. Minus six. Oops. I think I missed this one up. That needs to be a 20 four times five is 20 not 40. I'm six over 15. 90 minus 28. 70 on a six. A 64 all over 15. So the volume of the curve established by X squared. What while equals one bounded by the revolution around y equals negative. One is going to be 64 by 15. There you have it. Thank you.

So today we're going to be using integration. Solve questions on solids of revolution. So really problem on the volume of the solid generated by revolving the region pounded by the proble y equals X squared and the line y equals one will first start with the line y equals one is the axis of revolution. So the first step is to draw the curve. I'm gonna start by drawing y equals X squared. I like Teoh Just all the points on the plot. So go 0011 24 Then the same for the negatives. Negative 11 negative. 24 See how good I can draw this curve Now you and not too bad there. And then Michael's one. That one's a little bit easier for me. All right, so we got Michael's one here and in blue. We have y equals X squared. Okay, so the region bounded by That's gonna be in red, which is this region here. So the axis of revolution here is going to define which method we used to do the integration. So the fact that y equals one is one of the bounds of these curves that's actually going to be that same plot and green. So since the access of revolution is one of the bounding curves, that means that we're going to be using the disc method already. So after we've grown, the curves were determined. The radius direction. Here the radius direction is always going to be the opposite or perpendicular to the axis of rotation. So your access of rotations here it's parallel to the X axis. So we're going to have radi I and the extraction. Okay, so when you're doing the dis method, you got to think of this bounded curve as an infinite some of all of these very, very small disks. That's what the disc method is. So if you think of this curve here that we have, imagine that's the same one. We have all these very small, infinite discs that were going to some across the entire curd. So, as you can see, this radius is going to change. It's going to be a function of X, so you want to determine what that radius is in order to solve your volume function again for the disk method, while you it's going to be the integral oven area, some Dover height here. We're going to find the area of each of these little discs as we go from left to right. And your height of each disc is going to be that differential X. This is your differential here. I know them when you're finding the radius, that's going to just be the bounds of the two curves. So as you go from left to right, you have y equals one, and then you have a curve X squared. So your radius always going to be the top curve minus the bottom curve. So we're gonna have R of X because the radius says we said earlier, is a function of, well, your position on the X axis. It's gonna be the top curve, which is why equals one subtracted from the bottom curve, which is X squared, us going back. Teoh geometry area of a circle, Mr Find is hi or squared. We'll hear now are our radius isn't constant anymore. It's ah, it's a function of X. We're gonna have to square this function here, So r of X squared is going to be one minus x squared squared. That's going to equal one minus two x squared plus X to the fourth. Back it out to give me a little bit more space? No. So to solve this ah disc method integration here, we're gonna say the volume of this solid revolved about the curve y equals one is equal toothy some of all of these areas across this curve. So it's gonna be an integral to define your limits of integration. You're gonna find the two intersection points on this curve, and this one's a little bit simple or some might be a little bit more complex and require some actual algebra. But the two intersections or the points Ah, 11 and then negative 11 Since we're integrating in the X direction, we're going to use the X coordinate. So that's gonna be the integral from negative one toe one of your area function in your areas that pi r squared to save pine, then R squared. It's going to be one minus to x squared plus X to the fourth. Let's make sure that four looks a little bit better, and then area times your height, which is your differential X. Since, uh, pie is a constant, we're gonna pull that out same time is performing or integration Something to say that that's pie. We're gonna city X minus 2/3 X cubed plus 1/5 X to the fifth, evaluated from one to negative one. So when you go ahead and apply your confidence of integration, you have time, Times it could be one minus 2/3 plus one of his Not to make sure that fifth looks good subtracted friend that we're putting the negative one. So that's one plus 2/3 minus 1/5. Simplifying here. Pie, Uh, minus minus is plus there. So that's two minus 4/3. Do you think this getting a common denominator here that don't get us to pie? Times I always come to dominate. Here is 15. I have 30 minus 20 plus six all over 15 simplifying here, and that's give us 10 16/15. So that 16 hi over 15 volume of that curve is going to be a 16 pie over 15 that sets curve about the line like was one. Let's go to the next one for us. Gone ahead and drawn the curves for us this time just to kind of speed up this process and our new line of revolution is Y equals two. So since your axis isn't actually bounded by that curve, you're gonna have this space here that is empty. So if you can think about this, your new curve is gonna look something like that where this is also part of its He have this inner radius. It doesn't have any solid, so that can't go to the volume. So this man tha that we're gonna look at is the washer method. Any time that you have a void in your, uh, solid of revolution, you're going to use the washer instead of the disc method. So very similar formula. You know the volume is going to be pie R squared minus the smaller inner radius. Times I hear it's gonna be that differential again. So your two curves here bounded by big or which is your bigger radius, which is from your access of revolution to your bottom curve. The actual Trauger here in black and then your inner radius is going to be the smaller curve here. That inner radius is actually going to be a constant because it's going from the curve Y equals two twi calls one so Let's go ahead and get these two curves evaluated. So big are I want to be a function of X that's going to be top curve minus bottom curve. So that's why equals two minus curve X squared that's too subtract from X squared, then a little arm. It's actually not a function of X. It's just function. It's constant, So we're going to say R equals. Why equals two minus one, which he calls one. So let's go ahead and square these two. The inner radius is a little bit simpler. R squared is just gonna be one square, which is one R squared of X going vehicle for minus four X squared plus X to the fourth Again, these air, the infinite sums as you go from your same constant of integration here, that's gonna be negative. 1 to 1 again. That was found in the previous problem. A. By looking at the intersection of two curves y equals X squared, Michael's one. I swear to find all of these infinite, uh, infinitely thin washers. You're going to some of the areas of each of these to establish the volume That's the same. Was looking at washer here we're gonna rotate that access that would look like Yes, where your radius on the inside here is going to be a constant one in your larger radius is going to be a function of X established by the curve two months X squared. Let's go ahead and work through this problem. So the volume it's going to be the in a girl from negative one toe, one your area times by cherry There by your height, so R squared minus little R squared is going to be for minus one is three minus for X squared plus X to the fourth the X Applying the and girl here. We're going to say that that IHS bring out that pie Constant three x asked for thirds X cubed plus 1/5 X to the fifth Pretty sorry five and then your costs integration gonna be applied over here. One a negative one. So now going ahead and applying these volume was going to be pie. Applying one is going to be three minus 4/3 plus 1/5. You're gonna subtract by the other B minus three. Get a one here is going to be 4/3 B minus 1/5. Gonna go ahead and apply the signs. Simple. Fine. We have by six minus third's plus 2/5. Simplifying Begin. Please come. Denominators could be 15 3.5 by times. Six times 15 90 my ass. 40 plus six alone over 15 which simplifies down to 56. We were 15. So your volume of this bounded curve revolved about the line y equals one. This 56 by over 15. All right, last one. Now we're gonna go about the curve. Michael's one. We're sorry. They curves like als X squared and y equals one. The bounded region between these two revolt about y equals negative one. I've gone ahead and drawn your bounty curves as well as the axis of rotation here. So if you look at it this way, you gonna have another curve bound over here who hadn't draw this in for you? For the curve on the other side of the access of revolution, you have this well, hourglass shape here, a void in the middle. It's a little bit different from the last one. Uh, same method again, because your access of revolution is not one of the curves bounded by the solid revolution. We're going to use the washer method again. All right, So same formula that we've used all along here volume is going to be the area times. By your height since its differentials, we're gonna have the infinite testimony. Small I be your differential on X s. So let's go ahead and find your to, uh, radius curves. So big are it's going to be a constant here. It's now your talk her to your axes. Revolution is actually constant. That's your big are a little r. I know. I've switched up the colors on us. Now it's going to be that inner radius here. Your inner radius is actually the function of X on this plot. So let's go ahead and get big. Are since its constant here, you constant radius is going to be to we're sorry not to. You're curve is going to be one finest negative one, which equals two your inner radius. Little are it's going to be a function, uh, ex. That's gonna be a top curve minus bottom curve. That's going to be X squared minus negative one. So minus negative one, it's going to be X squared plus one it's going ahead and doing our squares here. R Squared of X It's going to be X to the fourth plus two x squared, plus one big R squared going to be two squared just for supplying our volume formula for the washer method. The is the integral from negative one toe. One constants haven't changed throughout this problem because it's been the same bounded region. So you're bounded regions going to be the intersection of the two curves, which is negative 1 to 1 times by your area, which is going to be I times big are squared, which is for minus little r squared. That's going to be minus X to the fourth Linus two X squared minus one time Sure. Infinitely small high d X trying to simplify here for me we have Hi. That's gonna be three in this two x squared. And that's next to the fourth DX. Go ahead and apply my integral. It's going to equal Pine times three X Here we go. Yes, 2/3 X cubed minus 1/5 extra Fifth applied at one negative one. Back it up a little bit So that's going to be Hi, I'm spy three 2/3 my ass. 1/5 subtracted from minus three. You have a negative. Negative is a positive. So that's plus 2/3 negative negatives. Another positive. That's plus 1/5 simplifying. We have pie. Go ahead and apply the negatives here. That's plus plus minus minus. Schooley. Positive. Six Negative. 2/3. Minus 2/3. Minus 4/3. Minus 1/5 plus negative. 1/5 is minus 2/5. Simplifying here. This is similar to the previous woman going to be. I was coming down. Mayors 15 again. Yeah, 90 minus 40. Minus six. Looks, I think I missed this one up that needs to be a 20 four times find is 20 not 40. I'm six over 15. 90 minus 28. 70 on a six is 64 all over 15. So the volume of the curve established by X squared. What while equals one bounded by the revolution around y equals negative. One is going to be 64 by 15. There you have it. Thank you.

Okay. What we want to do is we want to walk through the process to be able to develop a volume of a solid that is generated Yeah, by revolving some region about and we're going to actually do it for different um different axes, different lines. Okay. And so the solid that we're going to be working with is bounded. That region is bounded. Bye. Why equal to X squared? And why equal to one? Okay. And so let's, first of all, let's go ahead and get that that region drawn, I guess. Let's get that picture. Let's get a picture drawn in our head about that region. So we know this is a parabola that opens upwards. Um and so if I just kind of get that drawn and then Michael Dewan, is this horizontal line right here? Okay, Okay, so here is the region, this region right here is what we are going to be revolving. Okay, And the first time we're going to do it is we're going to revolve about um the line why equal to one. Okay, so we're gonna actually be taking this and revolving it about this line right here. Okay, so when I do that, you notice I kind of just get some kind of solid, maybe football looking thing, um something like that, it's some kind of football looking object or um I don't know, a solid donate or something like that. So sometimes these are kind of hard to imagine. So don't forget that. I'm really, when I revolve this, I am developing um cylinders um and just kind of multiple cylinders and then I'm summing them together. So the volume, so first of all I need to find the radius um and the radius is actually going to be going from my axis of rotation to the curve. Okay, and so that radius with respect to X is equal to one minus X squared because it goes from it's a link to that segment and it goes from 12 X squared. Ok so now the volume is equal to pi Integral and since I'm working with the XI'm going in from negative one 21 Alright. Somewhere from here. From negative 1 to 1 and it's r squared to one minus I X squared squared with respect X. Okay, so my volume is going to be equal to pi Integral from negative 1 to 1 of one minus two X squared Plus X to the 4th with respect X. Okay. So now I can just go ahead and integrate because we know how to do that. So this is going to be x -2/3 x cubed Plus 1/5 x to the 5th. And I'm going to evaluate at one and -1. So this is going to be equal to pi We have 1 -2/3 plus 1/5 minus all of negative one plus two thirds minus 1/5. And so when I combine all those numbers, we actually get 16/15 times five. Okay, so there is the volume of that first solid That is generated by revolving that region about the line. Why Gold one. And now the second time we do this we're going to be looking at the same region. But now we're going to revolve it about the line why equal to two? So what I'm going to be doing now my race this and kind of erase this. So now I'm going to have a line up here at why equal 2 2. So this is my line Why equal to two And I'm evolving it about that line right there. Okay, so now I noticed that I have between my region and my axis of rotation, I have this big gap right here, this big gap right here right here. And so that means when I revolve that region, my solid actually will have a hole in it. So now I know I'm going to have to raid I because I have an outside raid I that goes from the axis of rotation to the outside of the region and so that Is two -X. Squared. And then I'm gonna have a instance a smaller radius and that is going to go from uh huh. I didn't want to draw that there, that is actually going to go from my access of rotation to the inside of that region. And so that is that Radius or that length is 2 -1, which is one. Okay, so now what I have here Is the volume now is pie and I'm still going from negative 1-1 and it's going to be the big radius squared. So we have two minus X squared squared minus that smaller radius squared because I'm taking that reaching out of out of my solid. So this is going to be pie integral from negative 1-1 of four minus four X squared plus X to the fourth minus one and still integrating with respect X. So this will be equal to pi integral from negative 1 to 1 of three minus four X squared plus X to the fourth D X. Which is pi integral three is three x -4/3 x cubed Plus 1/5 x to the 5th. And I'm going to evaluate it at the upper limit of one and the lower limit of negative one. So this will be equal to pi We have 3 -4/3, Have 3 -4/3 um Plus 1/5 minus all of negative three plus four thirds minus once it. And then when I combine all of my light terms and common denominator on my fraction I get 56/15. Hi. And so there is my volume now of that solid that is generated by revolving that same region. Not about why I called the one but why equal to two. Okay, so very last time that we're going to do this and so to keep me from not scrolling, let's go ahead and get that picture drawn again. And so we have 111 and it's this region right here. Okay Um and now what we're gonna do is we were involved about why equal to negative one. So I actually get down here at -1. Have this line right here and we're going to revolve it about this line. And so you notice when I do that I once again develop a solid that has this whole because I have this gap right here between my region and my axis of rotation. Okay, so um I'm going to go ahead and now I'm gonna have to raid, I it goes from um the outside of my region to the axis of rotation, so that is going to be that raid i is one minus a negative one. So that gives me a length of two and the inside radius that I'm going to be taken away goes from the inside of the curve to that axis of rotation, so that is going to be X squared um minus a negative one, so plus they want, okay, so now my volume is equal to pi And I'm still going from negative 1-1 because that is the excesses that contain my region of my outside radius squared minus my inside radius squared. And I'm integrating still with respect to X because I'm rotating horizontally. And so this is going to be pie integral from negative 1-1, uh four -X to the 4th -2, x squared minus one. And so this will be equal to pi integral from negative 1 to 1 of three minus two, X squared -X to the 4th dx. So this is pie Integral of three is 3 x integral of two, X squared is two thirds X cubed. And the integral of X to the fourth is 1/5 X to the fifth. And once again I'm being evaluated at the upper limit of a plus one And the lower limit of a -1. And so this will be pie, This will be 3 -2/3 minus 1/5 minus all of negative three plus two thirds plus 1/5. And then when I combine all of my light terms, I get 64 over 15 pipe and there is the volume of that solid. I hope this helped um You understand um the difference of the different shapes and solids that are created and their volumes when you evolve a region about different things.


Similar Solved Questions

3 answers
E/e/E/8/7 1 1 WIde HU 6 i E7 18 L d J 2 5 I 2 I 2 U W 3 7 li Ti 2 5 ; 3 2 0 { VL 2 3 0 {8} 1 2 1 3 Vi [ 8 6 H { 1 { [ 1 8 3 L 3 0 L 3 [
e/e/E/8/7 1 1 WIde HU 6 i E7 18 L d J 2 5 I 2 I 2 U W 3 7 li Ti 2 5 ; 3 2 0 { VL 2 3 0 {8} 1 2 1 3 Vi [ 8 6 H { 1 { [ 1 8 3 L 3 0 L 3 [...
5 answers
10.A safety regulation states that the maximum angle of elevation for a rescue ladder is 720 . Include a well-labeled drawing: (round answers to the tenths place)Ifa fire department's longest ladder is 110 feet, what is the maximum safe rescue height? Sin728 Mellos 072 n?104_ LZfr. b.How far away from the building is the base of the ladder?
10.A safety regulation states that the maximum angle of elevation for a rescue ladder is 720 . Include a well-labeled drawing: (round answers to the tenths place) Ifa fire department's longest ladder is 110 feet, what is the maximum safe rescue height? Sin728 Mellos 072 n?104_ LZfr. b.How far a...
5 answers
Problem 9. Calculate the following = derivatives. Simplify as much as possible: a) Y'(x)= ? y =x sin * ;a"y 6) Find dm if y=sinx
Problem 9. Calculate the following = derivatives. Simplify as much as possible: a) Y'(x)= ? y =x sin * ; a"y 6) Find dm if y=sinx...
5 answers
Carbonic anhydrase is found in high concentration in(a) Leucocytes(b) Blood plasma(c) Erythrocytes(d) Lymphocytes
Carbonic anhydrase is found in high concentration in (a) Leucocytes (b) Blood plasma (c) Erythrocytes (d) Lymphocytes...
5 answers
Calculate $frac{partial f}{partial x}, frac{partial f}{partial y},left.frac{partial f}{partial x}ight|_{(1,-1)}$, and $left.frac{partial f}{partial y}ight|_{(1,-1)}$ when defined. HINT [See Quick Examples page 1098.]$$f(x, y)=x^{4} y^{2}-x$$
Calculate $frac{partial f}{partial x}, frac{partial f}{partial y},left.frac{partial f}{partial x} ight|_{(1,-1)}$, and $left.frac{partial f}{partial y} ight|_{(1,-1)}$ when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=x^{4} y^{2}-x $$...
5 answers
An object leaves the point $(0,0,1)$ with initial velocity $mathbf{v}_{0}=2 mathbf{i}+3 mathbf{k}$. Thereafter it is subject only to the force of gravity. Find a formula for the position of the object at any time $t>0 .$ Use feet and seconds.
An object leaves the point $(0,0,1)$ with initial velocity $mathbf{v}_{0}=2 mathbf{i}+3 mathbf{k}$. Thereafter it is subject only to the force of gravity. Find a formula for the position of the object at any time $t>0 .$ Use feet and seconds....
3 answers
Draw a Cayley diagraph for Z; using generators {2.5} =
Draw a Cayley diagraph for Z; using generators {2.5} =...
5 answers
(3x 3iy) dz; C is the contour in the figure shown below_
(3x 3iy) dz; C is the contour in the figure shown below_...
5 answers
QuCSUOHSuppose the supply function for units of a product is given by S (~) 1.252" | 5I Find the producers surplus ifthe equilibrium price is SSO. 0-S54.7805326.770554.7805136.550S176.89S95.1[None of the above
QuCSUOH Suppose the supply function for units of a product is given by S (~) 1.252" | 5I Find the producers surplus ifthe equilibrium price is SSO. 0-S54.78 05326.77 0554.78 05136.55 0S176.89 S95.1[ None of the above...
1 answers
Simplify each radical. Assume that all variables represent positive real mumbers. See Example 3 . $$ \sqrt[5]{\frac{1}{x^{15}}} $$
Simplify each radical. Assume that all variables represent positive real mumbers. See Example 3 . $$ \sqrt[5]{\frac{1}{x^{15}}} $$...
5 answers
Question 8 (4 points) An Sv2 reaction and its transition state are shown What are the partial charges on the Br and 0 atoms respectively?BrHSCh CHAHZCHatcitc HzoCHZCHZCH;Oa undThere should be no partial charges.Oxand &*O. and &"Oa" and &"
Question 8 (4 points) An Sv2 reaction and its transition state are shown What are the partial charges on the Br and 0 atoms respectively? Br HSCh CHAHZC Hatcitc Hzo CHZCHZCH; Oa und There should be no partial charges. Oxand &* O. and &" Oa" and &"...
5 answers
8 Solve using Green's function_y" _ 4y = &2ry(0) = 0, y(0) = 0.
8 Solve using Green's function_ y" _ 4y = &2r y(0) = 0, y(0) = 0....
5 answers
Given Cosl and Revenue functions C(Q) = q 8q2 55q 5000 and R(q) =3q" 2500q, what is Ihe marginal prolit at a produclion level of 40 iems?The marginal proli ISdollars per ilem
Given Cosl and Revenue functions C(Q) = q 8q2 55q 5000 and R(q) = 3q" 2500q, what is Ihe marginal prolit at a produclion level of 40 iems? The marginal proli IS dollars per ilem...
5 answers
Fx fy which corresponds to fyT f Lo"1
fx fy which corresponds to fy T f Lo " 1...
5 answers
Which of the following statements correctly describes cis-trans isomers?They have variations in arrangement around double bond_They have an asymmetric carbon that makes them mirror images.They have different molecular formulas:They have the same chemical properties.Question (1 point) polysaccharide you are studying contains unbranched (beta) glucosc molecules and cannot be digested by humans Which polysaccharide are your studying?DNAChitinStarchGlycogenCellulose
Which of the following statements correctly describes cis-trans isomers? They have variations in arrangement around double bond_ They have an asymmetric carbon that makes them mirror images. They have different molecular formulas: They have the same chemical properties. Question (1 point) polysaccha...

-- 0.022550--