Volume of the intersection of two cylinders by polar co-ordinates

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Homework Help Overview

The discussion revolves around finding the volume of the intersection of two cylinders using cylindrical coordinates. Participants are examining the integration limits and the geometric interpretations involved in the problem.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning the choice of integration limits, particularly the upper limit for dz. There is a discussion about the relationship between the coordinates and the geometry of the cylinders in the first octant.

Discussion Status

The conversation is ongoing, with participants providing comments and raising questions about the assumptions made in the original poster's approach. There is an exploration of different interpretations regarding the limits of integration.

Contextual Notes

Some participants note potential trivial mistakes in the original calculations, and there is a focus on the specific conditions under which the equations hold true, particularly in relation to the intersection of the cylinders.

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Volume of the intersection of two cylinders by cylinderical co-ordinates

Homework Statement




find Volume of the intersection of two cylinders by cylindrical co-ordinates


The Attempt at a Solution



IN the attached file I found it's 8(a^3)/3
It should be 16 not 8
 

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Last edited:
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I know that the mistake may be trivial but can anyone give me any comment!1
 
Why do you think the upper limit of the dz integration is r*cos(theta)? Don't you have z=sqrt(a^2-y^2)=sqrt(a^2-r^2*sin(theta)^2)?
 
Last edited:
Dick said:
Why do you think the upper limit of the dz integration is r*cos(theta)? Don't you have z=sqrt(a^2-y^2)=sqrt(a^2-r^2*sin(theta)^2)?

But in the first octent
x^2 + y^2 = r^2
y^2 + z^2 = r^2

so z=x=rcos(theta)

Isn't it?
 
That's only true along the curve where the two cylinders intersect. It's not true everywhere on the surface in the first octant.
 
Dick said:
That's only true along the curve where the two cylinders intersect. It's not true everywhere on the surface in the first octant.


Thanks Thanks Thanks
 

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