Confusion on the Volumes of Solids of Revolution

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Discussion Overview

The discussion centers around the confusion regarding the calculation of volumes of solids of revolution, specifically questioning the use of the average value of a function in this context. Participants explore the implications of averaging and the geometric interpretations involved in the volume calculation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions why the average value of the function cannot be used to determine the volume of a solid of revolution, providing an example with the function y=√x.
  • Another participant suggests that the term "average radius" may be misleading, noting that while the average height is 2/3, the elements contributing to the volume are not uniformly distributed in their distance from the x-axis.
  • A third participant references the theorem of Pappus, which relates the volume of a solid of revolution to the area of the region revolved and the distance traveled by the centroid, implying a connection to the concept of averaging.
  • A later reply seeks clarification on the specifics of the rotation about y=2/3, questioning the range of x being rotated in that scenario.

Areas of Agreement / Disagreement

Participants express differing views on the concept of average radius and its applicability to the volume calculation, indicating that there is no consensus on the correct interpretation or method to resolve the confusion.

Contextual Notes

There are unresolved assumptions regarding the definitions of average values and the geometric interpretations of the solids being discussed. The discussion also highlights the need for clarity on the specific parameters of the volume calculation.

grafs50
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I've been trying to figure out why you can't use the average value of a function to determine the volume of a solid of revolution.

As an example:

Trying to find the volume of a solid of revolution on y=√x from 0 to 1 around the x-axis.
The definite integral is 2/3, which divided by one is still 2/3 so it is the average value.

Then I tried to use turn this into a problem of finding the volume of a solid of revolution around the y=2/3. a radius of 2/3 from 0 to 1. I Since the sides were a straight line I tried to solve for the volume of a cylinder with radius 2/3 and height.

But this didn't work. I've been wracking my brain for like half an hour trying to figure out why but I've got nothing. Can anyone explain to me why this doesn't work?

Thanks in advance.
 
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It depends what you mean by "average radius". The average height of your y=√x is 2/3, but if you consider the elements in the volume of revolution, that is not their average distance from x axis. There are relatively more elements further from the axis.
 
Perhaps you are thinking of the (second) "theorem of Pappus" (also called the "theorem of Guidinus") that the volume of a solid of revolution is equal to the area of the region revolved times the distance from traveled by the centroid of the region ("average"?).
 
Forget my previous response. I realize I may not have understood this part:
grafs50 said:
Then I tried to use turn this into a problem of finding the volume of a solid of revolution around the y=2/3. a radius of 2/3 from 0 to 1. I Since the sides were a straight line I tried to solve for the volume of a cylinder with radius 2/3 and height.
If it's a rotation about y=2/3, what range of x is being rotated?
 

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