Volume between sphere and outside cylinder.

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Homework Help Overview

The discussion revolves around calculating the volume bounded by a sphere defined by the equation x²+y²+z² = 4 and the exterior of a cylinder described by x²+y²=2x. Participants are exploring various methods to set up the integration for this volume calculation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss using different coordinate systems, such as cylindrical and spherical coordinates, to simplify the integration process. There are suggestions to compute the volumes of the sphere and cylinder separately and then subtract, as well as to consider the geometry of the shapes involved. Some participants propose taking slices of the cylinder orthogonal to the x-axis for analysis.

Discussion Status

The discussion is ongoing, with various approaches being considered. Some participants have provided guidance on how to visualize the problem and suggested plotting the shapes to better understand the integration limits. There is no explicit consensus on a single method, but multiple interpretations and strategies are being explored.

Contextual Notes

Participants are grappling with the complexity of the cylinder's shape and its intersection with the sphere, which affects the integration limits. There are also discussions about the implications of using different coordinate systems and the challenges that arise from the geometry of the problem.

  • #31
In cylindrical, you have z, r, theta. What will be your integration order?
As a check on the answer, numerically I get about 9.66 for the volume removed by the cylinder, leaving 23.85 in the sphere. Sound right? Still working on the analytic result using Cartesian.
 
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  • #32
haruspex said:
In cylindrical, you have z, r, theta. What will be your integration order?
As a check on the answer, numerically I get about 9.66 for the volume removed by the cylinder, leaving 23.85 in the sphere. Sound right? Still working on the analytic result using Cartesian.

I get 9.644049708034451 for the volume removed. So that sounds about right. I integrated dz first, dr second and dtheta third. peripatein had the right general idea in the first post, except that the r factor in the measure was left out, the z limits are wrong, the r limits are wrong and the theta limits are also wrong. It just needs to be fixed. But seriously, if you are still working on the result in Cartesian, I'd give it up unless it's become an obsession. It's HARD to do that way. It's not THAT much of a challenge in cylindrical.
 
Last edited:
  • #33
Dick said:
I get 9.644049708034451 for the volume removed. So that sounds about right. I integrated dz first, dr second and dtheta third. peripatein had the right general idea in the first post, except that the r factor in the measure was left out, the z limits are wrong, the r limits are wrong and the theta limits are also wrong. It just needs to be fixed. But seriously, if you are still working on the result in Cartesian, I'd give it up unless it's become an obsession. It's HARD to do that way. It's not THAT much of a challenge in cylindrical.
I came to the same conclusion :biggrin:. In the end, I did it with a single integration. In the z plane, the area to be removed is the intersection of two circles. That turns into the sum of two sectors minus the sum of two triangles, so I could write those straight down. Final result is the removal of 16π/3-64/9, leaving 16π/3+64/9. Neatest way to express it is that it leaves 16/9 in the hemisphere it lies in.
 

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