Find the volume of the solid of revolution, or state that it does not exist. #2

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Discussion Overview

The discussion revolves around finding the volume of a solid of revolution formed by revolving the region bounded by the function f(x) = 6(4-x)^(-1/3) and the x-axis on the interval [0,4) about the y-axis. Participants explore different methods for calculating this volume, including the shell method and the washer method, while also addressing the need for limits and substitutions in their calculations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about which method to use for finding the volume of the solid of revolution.
  • Another participant suggests using the shell method and asks for the formula for the volume of an arbitrary shell.
  • There is a repeated inquiry about confirming the use of the shell method and the formulation of the integral for volume calculation.
  • One participant notes that while the shell method is suitable, the washer method could also be used, emphasizing the importance of checking results with multiple methods.
  • A suggestion is made to use a substitution (u = 4 - x) to facilitate the integration process.
  • Participants discuss the need to adjust the limits of integration when applying the substitution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the preferred method for calculating the volume, with multiple views on the shell and washer methods being presented. The discussion remains unresolved regarding the final formulation of the integral and the correct approach to limits.

Contextual Notes

There are unresolved aspects regarding the application of limits and the specifics of the substitution process, as well as the implications of choosing between different methods for volume calculation.

abc1
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I'm having some trouble with this problem:

Find the volume of the solid of revolution, or state that it does not exist. The region bounded by f(x)= 6(4-x)^(-1/3) and the x-axis on the interval [0,4) is revolved avout the y-axis.

How would I be able to tell whether to use the shell, disk, or washer method?
 
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We can use a function plotter to see the graph on the indicated interval:

View attachment 2238

Looks to me like I would try the shell method. Can you state the volume of an arbitrary shell?
 

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How can I tell that I have to use the shell method?

Would the formula for this be 2pi times the integral from 0 to 4 of x* (6(4-x)^(-1/3) dx ?

So then, how would I take the limit of this? What test should I use?
 
abc said:
How can I tell that I have to use the shell method?

Would the formula for this be 2pi times the integral from 0 to 4 of x* (6(4-x)^(-1/3) dx ?

So then, how would I take the limit of this? What test should I use?

You don't have to use the shell method, but can you see that the shell method allows for the computation of the volume as a single integral whereas the washer method will require an integral to be added to a cylinder? Either way would work though. In fact, I recommend to students to use more than one method both as a check as for the practice.

I always like to start by computing an element of the volume and then integrating. It appears that you do have the correct integral representing the volume. I think first though, I would use the substitution:

$$u=4-x$$

What do you have now?
 
Would it be 2pi times the integral from 0 to 4 of x* -(6(u)^(-1/3) du ?
 
abc said:
Would it be 2pi times the integral from 0 to 4 of x* -(6(u)^(-1/3) du ?

You want to write that $x$ out front in terms of $u$...:D
 
so the limit as u approaches infinity from 4 to 0 of 2pi times the integral from 0 to 4 of (4-x)* -(6(u)^(-1/3) du since we have to rewrite the limits of integration as well because of the u-substitution?
 

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