Use the cylindrical coordinates to find the volume

In summary, The volume of the solid S bounded by z=x^2 + y^2 and z=12 - 2x^2 - 2y^2 can be found by using cylindrical coordinates and integrating with respect to z, r, and theta. The limits for the integration should be from 0 to 2 for r, 0 to 2pi for theta, and from r^2 to 12-2r^2 for z. This results in a volume of 24pi.
  • #1
hbomb
58
0
I'm having trouble figuring out this volume.

Use the cylindrical coordinates to find the volume of the solid S bounded by z=x^2 + y^2 and
z=12 - 2x^2 - 2y^2

I've included a pic of what I think the regions and the solid look like

http://img.photobucket.com/albums/v87/hbombblack/3dgraphcopy.jpg

I think this is what the graph looks like. I would appreciate it if some one could show the steps of getting the volume, cause I'm kinda confused. I have an idea of what needs to be done, I just want to see if I'm on the right track before I get started.
 
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  • #2
The integral you need to use will be of the form

[tex]\int \int \int r \ dz \ dr \ d \theta [/tex]

Choosing your upper and lower limits should not be too difficult.

Here's a hint: which function z(r, theta) is above the other?
 
  • #3
I know all about cylindrical coordinates and how I should integrate. My question is when I integrate r. There's a constant radius where the two functions meet. So my question is, do I make two separate integrals? One for the top of the volume in question and another for the bottom. I solved the two system of equations and I came up with x^2 + y^2=4, which means that the radius is 2 where these two functions intersect. So for the first volume for dr I would do from x^2 + y^2 to 2. And for the second volume for dr I would do from 2 to 12 - 2x^2 - 2y^2. Is this correct? Or can I just go from x^2 + y^2 to 12 - 2x^2 - 2y^2?
 
  • #4
I'm having a hard time understanding your problem, sorry. :frown:

However, I think you'll find it simpler to disregard x and y completely. That way, you can deal with z as a function of r and theta.

The first such function is

[tex]z=r^2[/tex]

the second

[tex]z=12-2{r^2}[/tex]

Envision the r, z plane. In that plane, those two functions should intersect at (2, 4).

In the integral I showed you, the first integration is over z, not r. Integrate from the lower z to the upper z.

After that, integrate over r to receive the area in the r, z plane, which can be further integrated over theta.

Am I making any sense?
 
  • #5
Yes, it does make sense. What should my limits be for r though? That's the problem I'm having. The radius changes except for where the two functions intersect.
 
  • #6
Integrate r from 0 to 2.

It's a measure of distance from the z axis, so the lowest value it assumes is 0. The highest, in this case, is the point where the surfaces closes together, 2.
 
  • #7
It's
[tex]V=\int_0^{2\pi}d\phi \int_0^{2}rdr \int_{r^{2}}^{12-2r^{2}}dz=...=24\pi[/tex]
Because the z=x^2+y^2 and z=12-2x^2-2y^2 intersect at r=2, and r increases from 0 to the maximum of 2.
 
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Related to Use the cylindrical coordinates to find the volume

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in mathematics and physics to locate points in three-dimensional space. They consist of a radial distance from a central axis, an angle measured from a fixed reference direction, and a height or distance from a fixed plane.

2. How do you convert from Cartesian coordinates to cylindrical coordinates?

To convert from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), you can use the following equations:

r = √(x² + y²)

θ = arctan(y/x)

z = z

3. How do you find the volume of a solid using cylindrical coordinates?

The formula for finding the volume of a solid using cylindrical coordinates is:

V = ∫∫∫ r dz dθ dr

This integrates the function r with respect to z and θ, and then integrates the resulting function with respect to r. The limits of integration will depend on the shape of the solid.

4. Can you use cylindrical coordinates to find the volume of any solid?

Yes, cylindrical coordinates can be used to find the volume of any solid that can be described using a cylindrical geometry, such as a cylinder, cone, or sphere. However, for more complex shapes, it may be easier to use other coordinate systems.

5. What are some advantages of using cylindrical coordinates to find volume?

One advantage of using cylindrical coordinates is that they can simplify the calculation of volume for certain shapes, such as cylinders and cones, which have a constant radius and height. Additionally, cylindrical coordinates can be useful for solving certain physical problems, such as those involving rotational symmetry.

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