Setting up a triple integral in cylindrical coordinates?

In summary, the conversation discusses finding the volume of material cut from a solid sphere and a cylinder. The sphere has a radius of 3 and is centered at the origin, while the cylinder has an equation in polar coordinates. The conversation also mentions converting to cylindrical coordinates and provides an integral with the correct limits for the problem.
  • #1
VinnyCee
489
0
The problem says to find the volume of material cut from the solid sphere,

[tex]r^2 + z^2 \le 9[/tex]

by the cylinder,

[tex]r = 3\sin\theta[/tex]

I don't know how to graph the first equation, but I can do the second in polar coordinates. How do I go about converting to use cylindrical coordinates?
 
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  • #2
VinnyCee said:
I don't know how to graph the first equation

It's a sphere of radius 3 centered at the origin:

[tex]r^2+z^2=x^2+y^2+z^2=9[/tex]


but I can do the second in polar coordinates. How do I go about converting to use cylindrical coordinates?

It's already in cylindrical coordinates. The equation is implying that [tex]r=3sin(\theta)[/tex] for all z.
 
  • #3
I did a crude sketch and came up with this integral

Is this correct limits for the problem?

[tex]\int_{0}^{2\pi}\int_{0}^{3\sin\theta}\int_{-\sqrt{9 - r^2}}^{\sqrt{9 - r^2}}\;dz\;r\;dr\;d\theta[/tex]
 
  • #4
"Cylindrical coordinates" is simply polar coordinates with "z" added.
[tex] r= 3 sin \theta[/tex], a circle with center at (0, 3/2), radius 3/2, in polar coordinates is a cylinder running parallel to the z axis in cylindrical coordinates.
Since r2= x2+ y2, the sphere, x2+ y2+ z2= 9 is r2+ z2= 9 in cylindrical coordinates.
 
  • #5
Looks good to me.
 
  • #6
Many Thanks

Thank you both. When I have more time on my hands, I will be sure and return the favor to someone here someday :smile:
 

Related to Setting up a triple integral in cylindrical coordinates?

1. What is the formula for setting up a triple integral in cylindrical coordinates?

The formula for setting up a triple integral in cylindrical coordinates is ∭f(r,θ,z) rdrdθdz. This means that the integral is evaluated over a volume in the shape of a cylinder, with the limits of integration being the radius (r), angle (θ), and height (z).

2. How is the region of integration determined in cylindrical coordinates?

The region of integration is determined by the limits of each variable. In cylindrical coordinates, the limits of r, θ, and z will determine the shape and size of the region being integrated over. These limits can be determined by the given boundaries of the problem or by visualizing the region in the cylindrical coordinate system.

3. What are the advantages of using cylindrical coordinates for triple integrals?

Cylindrical coordinates are advantageous for triple integrals because they are better suited for solving problems involving cylindrical symmetry or cylindrical objects. This coordinate system simplifies the integration process and can lead to easier and more efficient calculations.

4. Can cylindrical coordinates be converted to Cartesian coordinates?

Yes, cylindrical coordinates can be converted to Cartesian coordinates. The conversion formulas are x = rcos(θ), y = rsin(θ), and z = z. This means that any triple integral in cylindrical coordinates can be rewritten as a triple integral in Cartesian coordinates, and vice versa.

5. What are some real-world applications of cylindrical coordinates in triple integrals?

Cylindrical coordinates in triple integrals have various real-world applications, such as calculating the volume of a cylindrical tank, finding the mass of a cylindrical object, or determining the electric field around a cylindrical wire. They are also commonly used in physics, engineering, and other fields to solve problems involving cylindrical symmetry.

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