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

  • #1
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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  • #2
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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  • #3
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?
 
  • #4
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:



What do you have now?
 
  • #5
Would it be 2pi times the integral from 0 to 4 of x* -(6(u)^(-1/3) du ?
 
  • #6
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 out front in terms of ...:D
 
  • #7
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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