5

(1 point)Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves Y = the line y = 4.and y = 4 aboutVolume...

Question

(1 point)Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves Y = the line y = 4.and y = 4 aboutVolume

(1 point) Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves Y = the line y = 4. and y = 4 about Volume



Answers

$1 - 18$ Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$
y = \frac { 1 } { 4 } x ^ { 2 } , x = 2 , y = 0 ; \quad \text { about the } y
$$

Given a set of curves and the line were asked by the volume of the solid obtained by rotating the region bounded by these curves about this line. The curves include Why equals 1/4 squared X equals two and Y equals zero in the line that were rotating about is the y axis. First, let's graph this region more convenient to do this. Just include the first quadrants so graphing, why equals 1/4 x squared. We have points at the origin. We also have a point at to one and that one 1/4 take a shape like this. We have a line of X equals two, which is a vertical line in white zero. So it's this region in red that we're interested in finding the area. You're sorry rotating about the Y axis. To do this, I'm gonna need the 1st and 2nd quadrants. If you're doing this rotation, we obtain a solid. It looks like this. If we take a cross section, it looks something like this to get the cross section is in fact a washer with an inner radius which is rearranging our equation to root y and an outer radius, which is simply to therefore the area of the washer is pi Times two squared in a spy times to root y squared simplifies two pi times four minus four Why and therefore thieve volume of this solid is the integral from y equals zero toe one of the area A Y de y yeah, mhm. This is the anti girl from 01 of pi times four minus four Why de y and taking it head derivatives This is pi times or why minus two y squared 01 substituting This is pi times for minus two, which is simply two pi.

And this problem. We are talking about a direct application of integration and that is finding the volume under a curve. So how do we do this? Um, there are two methods. There's a cylindrical shell method and washer method. We're going to be using the washer method. And so this will make more sense when you talk about the function were given. So this star coordinate plain and this blue line is our function Thes dotted lines are the interval that we're investigating and this red line is essentially the height of our function. So now what we do is we essentially rotate this or reflected across the X axis. So you can imagine we have this really long line here and we essentially call it a disk washer and we essentially add up all of these washers together to find the volume that would be created by our function. So you can see that this is a direct application of integration. We're finding essentially a sum of all of these washers, which is what an integral will do. So now let's find the volume for the function were given. So the volume is going to be equal to pi times the integral from 1 to 4 of our radius times dx the radius again is this red line. Imagine that this would be a full circle. This would be the radius. So now we could plug in some information. Our volume would be equal to pi times the integral from 1 to 4 of one over X squared in d X then we can. And to derive this, our volume would be equal to pi times Negative one over X evaluated from 1 to 4. So our volume is pi times negative 1/4 plus 1/1 and we can simplify that to three pi over four. So what we just found is that using the washer method and the function were given the volume that would be created by that washer is through pi over four. So I hope this problem helped you understand a little bit more about how we can use the integral to find the volume under our function. Even specifically, how we go about, um, one of these problems using integration and a given function

Okay, let's draw this first. So we can see what area they're talking about that we're going to rotate. So Y equals X. That this green line? Because you're 001122. That's why we'll six Michael zero. That would be the X. Axis X equals two. Maybe this vertical line X equals four. That vertical line. So you can clearly see the area that we're talking about, the area and here. Okay, now we're gonna send it around the line. X equals one. That's kind of in the wrong place. Just a little bit. When I'm moving over, just a little bit, X equals one right here. So, we gotta cut a slice because we're going around a vertical line, we cut horizontally. Okay, when that sliver goes around, it's gonna leave a hole. Okay, get out some terrible picture. Okay? But when this sliver goes around, it's gonna leave a hole. So we have a washer. Okay, it looks like this. There's the line white goes one right through there. Where's the whole coming from? Well, it's from this blue line right here to when it rotates around and it ends up over here. Okay, So in between those two holes there is nothing I mean in between those two lines. Okay, what's the thing look like? Well, the bottom is flat and then the inside is kind of lower than the outside kind of like that. Okay. That's pretty weird. All right, so, to find the volume, it's uh pie big R squared minus little R squared H. Where h is the thickness of the disc or shell or washer which is deep y. Big R is the distance from the axis of rotation out to the big piece. All right so watch me draw it's from here to here. Okay well this much is one and from here to here is some X. Value. So it's X -1. Which ex well it's x equals 4 -1 which is three and that's constant throughout this whole shape. Okay. But then little R is a problem and here's the problem. Oops little or changes depending on if you're down here or if you're up here so down here little R. Is four minus two. We'll see at the bottom and then at the top little or is from here to there. So just some Why value Which Y? Well it's seeing the line Y equals X. I'm sorry some X I'm sorry it's some X value. Okay from here. From the Y axis out that's X. But you're going to integrate dy so you have to change it to Y but conveniently X is equal to Y. So I would have got that one. Right anyway. All right so here's what I got volume is somewhere to somewhere and we'll worry about that in a minute. Pie big R squared minus little R squared. Dy plus pie big R squared minus little R squared. Do you want? Okay so my picture is getting kind of messier. Let's see if I can get some of this out of here. It well pretty much takes the whole thing away erasers. Huh? I would have known. All right so yeah down here you can see that this is the distance between the two sides. Is too is too it's too it's too it's too it's too and then right here right here it stops being too. Well that's when the line x equals two hits the line, y equals x. So that's at y equals two. So it's like well 0 to y equals two And then y equals to two. Well it stops when it gets here which is when y equals X. Hits X equals four. So that's 4-4. All right so we got 0-2 pi 16 -4. Which is 12 do you I that's 12 pi. Why? From 0 to 2? That's 24 pi plus This one is 2-4 pi That's a 4 9 and nine. Almost made a big mistake there. 16 minus y squared. Dy Okay I forgot to integrate. Oh no I got to be in a great part here. It comes 16 y minus y cubed over three From 2- four. So 64 64 3rd minus 30 to minus eight thirds 64 -32. That's 32 minus 64 plus eight. That's 56 3rd and that's negative 56 3rd 32 times three. That's 96 96 -56. That's 43rd pie. Okay so 24 pi plus 43rd pie, 24 times three, plus 40. That's 72, 112 for three Pi. That's the volume.

You if I have the equations. Wyck was 1/4 X squared X equals two y equals zero, and we're going around the why access to first ball 1/4 x squared is a parabola, and it's like a shortened proble. X equals two would be a vertical line over where access to and why why equals zero is the X axis. So the area that has been bounded by those equations is this part of just colored in with that mind. If I'm rotating around the Y axis, then I do see that there's this missing space. As I rotate that top piece look at all this distance. It has to cover where there's nothing filled in. So that means this is a washer set up and for the washer set up, volume is pi times the integral capital R squared minus little R squared. You want to use whatever variable is your excess or parallel to your access. So in this case, D Y for the Y axis, and that's also gonna tell me my bounds. Why wanted Why, too? The bounds for this area are why equals zero that's given as that first equation, but for the 2nd 1 up here for why, too? It's like, what is this value? So we do know something about where that point happens. That point happens when X is too. So if I can use my equation for why knowing that X is too, then I can find that y value. So why you ghouls? 1/4 2 squared. Well, that's 1/4 of four. So my upper bound is gonna be one. So we're going from 0 to 1. We have capital R squared, minus little r squared. So now we need to consider what is capital art? That would be the radius if this whole thing were filled in and actually notice If you look at the bottom of the shaded area, that is the radius, if everything were fill in there is that one relief in infinite strip there that goes all the way from the access to the edge of our shape. And that's defined by the value of two. That vertical line back to the axis zero. So for capital are I'm gonna fill in two and then for lower case R. It's the stuff that I need to take away. So it's the stuff here in the middle that's missing notice. The stuff that's missing goes from the parabola back to the access. It goes from that curved shape back to the axis. Well, I can't fill that in because if I feel that in, I have the wrong variable. So I need to Saul to be in terms of Why. So let's solve this 1/4 ex squared to be in terms of why we're gonna have to multiply before over and then square root both sides. And I don't want the plus or minus here because obviously the exes, only on the right side of the origin it's only the positive version. So I don't need that plus or minus. So two squared of why is what I want to fill in here. So at this point, I'm ready to go ahead and solve this integral. I have four minus four. Why, if I square that second term so as I integrate, I get pi times the whole thing for why minus two y squared going from 0 to 1, filling in the upper bound of wine is four minus two You and then, if I feel in that lower bound of zero, it's nothing. So the simplified answer here is to buy for my volume


Similar Solved Questions

5 answers
SAMPLE MII Solve the diffcrential cquation f'(x) =-4x' + 3r- | given that ((I) = 0 Find thc indefinite intcgrals: fc-Sx+2+- dx fsec' 4x &x Jx#-3 & [x VI+; dr
SAMPLE MII Solve the diffcrential cquation f'(x) =-4x' + 3r- | given that ((I) = 0 Find thc indefinite intcgrals: fc-Sx+2+- dx fsec' 4x &x Jx#-3 & [x VI+; dr...
5 answers
Lim 2sin2e 10) 0-0 30
lim 2sin2e 10) 0-0 30...
5 answers
Which structures are those compounds that contain a benzene ring and have molecular ions in the mass spectrum at m/z = 132?NCNHzCN
Which structures are those compounds that contain a benzene ring and have molecular ions in the mass spectrum at m/z = 132? NC NHz CN...
5 answers
What is the percent ionization of HzNNHz in a solution with a concentration of a 0.210 M? (Kb = 1.3 * 10-6)
What is the percent ionization of HzNNHz in a solution with a concentration of a 0.210 M? (Kb = 1.3 * 10-6)...
5 answers
Suctching frequencies for the following Lypes of bonds: Specity the region of characteristic RO-H RO R. Aad aeometc ~Ibonds (Tpls /
Suctching frequencies for the following Lypes of bonds: Specity the region of characteristic RO-H RO R. Aad aeometc ~Ibonds (Tpls /...
3 answers
Find the eigenvalues and corresponding eigenvectors for the matrix Is the matrix A diagonalizable? Ifit is, what is the matrix P that diagonalizes A and what is D the diagonal matrix that is similar t0 A?
Find the eigenvalues and corresponding eigenvectors for the matrix Is the matrix A diagonalizable? Ifit is, what is the matrix P that diagonalizes A and what is D the diagonal matrix that is similar t0 A?...
5 answers
Determine the Ka of an acid whose 0.527 M solution has a pH of 2.45.
Determine the Ka of an acid whose 0.527 M solution has a pH of 2.45....
5 answers
Calculate the work (in joules) required to pump all of the water out of the tank. Assume that the tank is full, distances are measured in meters, and the density of water is $1.000 mathrm{~kg} / mathrm{m}^{3}$.The hemisphere in Figure 8; water exits from the spout as shown.
Calculate the work (in joules) required to pump all of the water out of the tank. Assume that the tank is full, distances are measured in meters, and the density of water is $1.000 mathrm{~kg} / mathrm{m}^{3}$. The hemisphere in Figure 8; water exits from the spout as shown....
5 answers
Di Mvhitedeke temales anhaerken Iseal Imatel [email protected] Hakankh Mseel HhbieU EEra MR
Di Mvhitedeke temales anhaerken Iseal Imatel [email protected] Hakankh Mseel HhbieU EEra MR...
5 answers
Draw the structure of 1 and write the appropriate reagentls (A- E) for each reaction (one step reaction) along the arrow: (12)CH,OH
Draw the structure of 1 and write the appropriate reagentls (A- E) for each reaction (one step reaction) along the arrow: (12) CH,OH...
5 answers
Insulation According to the figure in Exercise $81,$ which of the materials is the best insulator? the poorest? Explain.
Insulation According to the figure in Exercise $81,$ which of the materials is the best insulator? the poorest? Explain....
5 answers
A cube exists with sides X by X by X. If the dimensions aredoubled, what happens to the new are of the cube?
A cube exists with sides X by X by X. If the dimensions are doubled, what happens to the new are of the cube?...
5 answers
22. (a The number a is called a double root of the polynomial function f if f(x) = (x a)2g(x) for some polynomial function g Prove that a is double root of f if and only if is a root of both f and f' (6) When does f (x) = ax2_ +bx +c (a # 0) have double root? What does the condition say geometrically?
22. (a The number a is called a double root of the polynomial function f if f(x) = (x a)2g(x) for some polynomial function g Prove that a is double root of f if and only if is a root of both f and f' (6) When does f (x) = ax2_ +bx +c (a # 0) have double root? What does the condition say geometr...
5 answers
Evaluate the function for the given value ofxglx) = 4x - 3,hlx) = 4Vr-T, (h 0 g)(-2) = ?161Unclchned 16v
Evaluate the function for the given value ofx glx) = 4x - 3,hlx) = 4Vr-T, (h 0 g)(-2) = ? 161 Unclchned 16v...
5 answers
Particle moves along a straight line and its position at time t is given by s(t) 2t3 27t2 + 108t where s is measured in feet and t in seconds.Find the velocity (in ftlsec) of the particle at time t 0The particle stops moving (i.e. is in a rest) twice, first when t =and again when tWhat is the position of the particle at time 18?Finally, what is the TOTAL distance the particle travels between time 0 and time 18? ft
particle moves along a straight line and its position at time t is given by s(t) 2t3 27t2 + 108t where s is measured in feet and t in seconds. Find the velocity (in ftlsec) of the particle at time t 0 The particle stops moving (i.e. is in a rest) twice, first when t = and again when t What is the po...

-- 0.021229--