Finding this volume without cylindrical shells

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SUMMARY

The discussion revolves around calculating the volume of a region bounded by the curve y=1/(x^2), the vertical lines x=e and x=e^3, and the x-axis, when rotated around the y-axis. The original method employed was the cylindrical shells technique, yielding a volume of 4π. The user inquired whether it is feasible to solve this problem without using cylindrical shells but ultimately resolved the issue independently.

PREREQUISITES
  • Understanding of volume calculation methods in calculus
  • Familiarity with the cylindrical shells method
  • Knowledge of the properties of the function y=1/(x^2)
  • Concept of rotating regions around axes in integral calculus
NEXT STEPS
  • Explore alternative volume calculation methods such as the disk method
  • Study the application of integration techniques in finding volumes of revolution
  • Investigate the properties of the function y=1/(x^2) in different contexts
  • Practice solving similar problems involving rotation around the y-axis
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Students and educators in calculus, mathematicians interested in volume calculations, and anyone seeking to deepen their understanding of integration techniques in geometry.

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This is not a homework problem this is just a problem I was thinking about whether or not it would be possible to solve without cylinderical shells

The region bounded by y=1/(x^2) x=e x=e^3 and the x-axis Rotated around y axis.. I did cylindrical shells and got 4pi and then wondered if I could do without.

I have been trying to figure out how to do it for about an hour but can't come up with anything that works.

Is it possible to do by hand without cylinderical shells? If so I would like so advice on how to go about it.
 
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