Calculating Volume from Revolved Area

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Discussion Overview

The discussion revolves around the calculation of volume from the area of an arbitrary shape when revolved around an axis, specifically the y-axis. Participants explore theoretical aspects and mathematical principles related to solids of revolution.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant recalls using integration to calculate volume by revolving shapes and questions if volume can be determined from the area of a shape without a defined equation.
  • Another participant argues that volume cannot be directly calculated from area alone, citing the example of a sphere and a cylinder with the same cross-sectional area, emphasizing that the integral of the square of a function is not a fixed multiple of the integral of the function itself.
  • A participant mentions a theorem related to the volume of a solid of rotation being equal to the area of the region rotated multiplied by the circumference of the circle swept out by the centroid, noting the challenge of determining the circumference without the boundary formula.
  • Further clarification is provided regarding the centroid's distance and its relation to volume, with a participant recalling "Pappus' Theorem" as relevant to the discussion.
  • There is a light-hearted exchange regarding the name "Pappus' Theorem," with participants noting its dual references in geometry.

Areas of Agreement / Disagreement

Participants express differing views on the ability to calculate volume from area alone, with some supporting the idea that additional information is necessary, while others reference relevant theorems that may provide insight into the relationship between area and volume.

Contextual Notes

Participants acknowledge the limitations of their knowledge regarding specific theorems and mathematical details, indicating that some assumptions and definitions may need clarification for a complete understanding.

Who May Find This Useful

This discussion may be of interest to students and professionals in mathematics and physics, particularly those exploring concepts of volume, area, and solids of revolution.

eberg
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I remember calculating volume by revolving shapes around axis using integration in school. I have an arbitrary shape shape that has no easily attainable equation, if I know the area of that shape, can I calculate the volume?

This is an example. I know this area, and want the volume if revolved around the y-axis:

http://img149.imageshack.us/img149/1/shapepo1.th.png

-Eric
 
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eberg said:
I have an arbitrary shape shape that has no easily attainable equation, if I know the area of that shape, can I calculate the volume?

Hi Eric! :smile:

No … compare the volume of a sphere with the volume of a cylinder with the same cross-section area.

Mathematically, volume = π∫(f(z))2dz, and area = ∫f(z)dz, and the integral of (f(z))2 isn't a fixed multiple of the integral of f(z). :smile:

hmm … how about the surface area of the solid? :wink:
 
OK, thanks for the response.
 
There is a theorem (I can't quite remember the name right now) that the volume of a solid of rotation is equal to the area of region rotated and the circumference of the circle swept out by the centroid of the region. Of course, without knowing the formula for the boundary of the region you can't find that circumference exactly but for reasonably shaped regions you might be able to approximate it by seeing where lines from opposite points intersect.
 
HallsofIvy said:
There is a theorem (I can't quite remember the name right now) that the volume of a solid of rotation is equal to the area of region rotated and the circumference of the circle swept out by the centroid of the region. Of course, without knowing the formula for the boundary of the region you can't find that circumference exactly but for reasonably shaped regions you might be able to approximate it by seeing where lines from opposite points intersect.

Hi HallsofIvy! :smile:

(i think you omitted the word "product" :wink:)

yes, the centroid would be at a distance of ((1/2)∫(f(z))2dz)/area,

so the volume is 2π times that times the area. :smile:
 
Thanks, tiny-tim.
And I now recall that it is "Pappus' Theorem".
 
good ol' Pappus of Alexandria …

HallsofIvy said:
Thanks, tiny-tim.
And I now recall that it is "Pappus' Theorem".

Hi HallsofIvy! :smile:

At first, I thought, what's he talking about :rolleyes: … isn't Pappus' theorem that projective hexagon thing?

Then I looked in http://en.wikipedia.org/wiki/Pappus'_Theorem#Theorems", and found that good ol' Pappus :smile: had two theorems …
Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.

so you're absolutely right! :biggrin:
 
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