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

  • #1
abc1
9
0
Find the volume of the solid of revolution, or state that it does not exist. The region bounded by f(x)= the square root of ((x+3)/(x^3)) and the x-axis on the interval [1,infinity) is revolved around the x-axis.

I tried using the disk method: pi* (sqrt(((x+3)/(x^3)))^2
Then I think I have to take the limit as b is approaching infinity from 1 to b of pi* (sqrt(((x+3)/(x^3)))^2. But I don't know how to take the limit now. Am I doing this problem correctly? Can someone please help me solve it?
 
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  • #2
We are given:



The volume of an arbitrary disk is:



where:



Hence:



And so:



Since this is an improper integral, we may write:



I would suggest rewriting the integrand:



Now, apply the FTOC and then take the limit of the result. Can you proceed?
 
  • #3
Thank you so much for replying! I was just wondering, would it be possible to use lhopital's rule to find the limit since V=πlimt→∞(∫t1x+3x3dx) would be infinity over infinity? I tried that and I got 1/(3x^2) and then tried to apply the fundamental theorem of calculus, but I got the wrong answer, and I don't understand why.
 
  • #4
Also, I tried proceeding from where you left off, applying the FTOC and I got pi * (lim as b approches infinity of (b^-2 +3b^-3) - 4. So then wouldn't that equal pi * ( infinity + 4) so it would be infinity so it would diverge?
 
  • #5
No, it's not an indeterminate form...I would write:





Can you take the limit now?
 
  • #6
Thanks so much! I got 5/2!

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I have one more question though. Why wasn't it an indeterminate form? It looked like it would be infinity over infinity.
 
  • #7
abc said:
Thanks so much! I got 5/2!

- - - Updated - - -

I have one more question though. Why wasn't it an indeterminate form? It looked like it would be infinity over infinity.

Don't forget the factor of . :D

Do you mean the integrand in its original form? The following is not true in general:

 
  • #8
Oh okay! Thanks so much again for your help! :)
 
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