Find Mass of Ball B w/ Triple Integrals & Cylindrical Coordinates

In summary, the conversation is about finding the mass of a ball B using cylindrical coordinates, where the density at any point is proportional to its distance from the z-axis. The triple integral of K*r^2dz*dr*dθ is used, with limits for the integral being -sqrt(a^2-r^2) to sqrt(a^2-r^2) with respect to z, 0 to a with respect to r, and 0 to 2*pi with respect to theta. The speaker is having difficulty integrating the function and asks for help, but the other person suggests using spherical coordinates instead.
  • #1
physicsss
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Find the mass of a ball B given by "x^2+y^2+z^2≤a^2" if the density at any point is proportional to its distance from the z-axis using cylindrical coordinates So is the density equal to K*sqrt(x^2+y^2), or K*r?


Using triple integral of f(rcosθ, rsinθ, z)*r*dz*dr*dθ) I got the following: the triple integral of K*r^2dz*dr*dθ, and the limits for the integral w/ respect to z being from -sqrt(a^2-x^2-y^2) to sqrt(a^2-x^2-y^2), which becomes -sqrt(a^2-r^2) to sqrt(a^2-r^2), limits w/ respect to r being from 0 to a and w/ respect to theta from 0 to 2*pi.

Doing it I find the integral very hard to integrate because I can't do u-subsititon with two r^2's.

Am I doing anythign wrong? Thanks in advance.
 
Last edited:
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  • #2
U can very well use spherical coordinates.I'm sure the integration is immediate.

Daniel.
 
  • #3
I can't. the stupid book asks us to use cylindrical coordinates.

Is my setup right though?
 
  • #4
Why?What's the point...?Why is it hard to integrate...

[tex] M=k\int_{0}^{2\pi}d\varphi \int_{0}^{a}r^{2} \ dr \int_{-\sqrt{a^{2}-r^{2}}}^{+\sqrt{a^{2}-r^{2}}} \ dz [/tex]

I think it's trivial.

Daniel.
 
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  • #5
You forgot a r...in cylindrical coordinates you there's an extra r.
 
  • #6
You're right,i edited.Then do it...It's not difficult.You can make a "sin/cos" substitution and solve it easily.

Daniel.
 

1. How do you calculate the mass of ball B using triple integrals and cylindrical coordinates?

To calculate the mass of ball B using triple integrals and cylindrical coordinates, we first need to determine the bounds of integration for the three variables (r, θ, and z). This can be done by setting up a cylindrical coordinate system with the center of the ball at the origin. Then, we can use the triple integral formula for mass: M = ∭ρ(r, θ, z) dV, where ρ is the density function of the ball.

2. What is the difference between using triple integrals and cylindrical coordinates compared to other methods for finding the mass of a ball?

Using triple integrals and cylindrical coordinates allows us to calculate the mass of a ball more efficiently and accurately compared to other methods. This is because cylindrical coordinates are well-suited for solving problems involving cylindrical objects, such as balls. Additionally, triple integrals allow us to integrate over three variables, which is necessary for calculating the mass of a three-dimensional object like a ball.

3. Can the mass of an irregularly shaped ball be calculated using this method?

Yes, the mass of an irregularly shaped ball can still be calculated using triple integrals and cylindrical coordinates. This method is not limited to only perfectly spherical balls, as long as the density function is known and the bounds of integration can be determined for the given shape.

4. Are there any real-world applications for finding the mass of a ball using triple integrals and cylindrical coordinates?

Yes, there are many real-world applications for this method. For example, in physics, calculating the mass of a planet or a spherical asteroid would require the use of triple integrals and cylindrical coordinates. In engineering, this method can be used to determine the mass of cylindrical objects, such as pipes or containers.

5. Is it possible to use a different coordinate system, such as spherical coordinates, to find the mass of a ball?

Yes, it is possible to use a different coordinate system to find the mass of a ball. However, cylindrical coordinates are the most commonly used coordinate system for solving problems involving cylindrical objects. Spherical coordinates may be used for more complex shapes, but the process for finding the mass would be similar to using cylindrical coordinates and triple integrals.

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