Volume bound by rho=2+2cos phi

In summary, the conversation discusses finding the volume bounded by rho=5+2cos(phi) using cylindrical coordinates and the volume differential dV=rho squared sin phi drho dphi d theta. The questioner is unsure of the limits for dtheta and dphi and asks for verification on their answer of 653.33 Pi.
  • #1
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Homework Statement



Find the volume bounded rho=5+2cosphi

Homework Equations


dV=rho squared drho d phi d theta



The Attempt at a Solution



I am guessing this is some cylindrical shape. Theta should be 0-2pi and phi=0 pi/2
 
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  • #2
You are missing a sine in your volume differential. Otherwise, what is the question? What work have you done on it so far?
 
  • #3


ok typo: dV= rho squared sin phi drho dphi d theta

I cannot visualize this object to find the volume bound by rho= 5+2cos(phi)

I want to know the limits of dtheta and dphi

Integrating rho squared is easy
I am guessing the limits are 0-pi/2 for phi
and 0-2pi for theta
 
  • #4
You will need to go all the way to pi for phi. Your theta bounds are correct. You don't need to worry about visualizing it - you know the volume differential:

[tex]\int dV = V[/tex]
 
  • #5
just to check answer: I got 653.33 Pi.

Can anyone verify if that is correct?
 

FAQ: Volume bound by rho=2+2cos phi

What does "rho=2+2cos phi" represent in terms of volume?

"rho=2+2cos phi" represents a three-dimensional geometric shape known as a torus, where the cross-section of the shape is a circle with a radius of 2 and the center of the circle moves along the x-axis according to the cosine of the angle phi.

How is the volume of a torus calculated?

The volume of a torus can be calculated using the formula V = π^2 * R * r^2, where R is the distance from the center of the torus to the center of the tube and r is the radius of the tube.

What is the range of possible values for the radius of the torus?

The radius of a torus can range from 0 to infinity, as long as the inner radius (r) is smaller than the outer radius (R) to create a hollow space within the shape.

Can the volume of a torus be negative?

No, the volume of a torus cannot be negative as it represents a physical amount of space and must have a positive value.

Are there any real-life applications of a torus with "rho=2+2cos phi"?

Yes, this shape can be seen in various objects such as doughnuts, lifebuoys, and tire tubes. It also has applications in fields such as architecture, engineering, and physics.

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