Rotation around a curve. Find the Volume.

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SUMMARY

The discussion centers on calculating the volume generated by rotating a function around its axis, specifically addressing the conditions required for such calculations. Key points include the necessity for the function to be continuous to ensure meaningful results. The conversation highlights that rotating a constant function does not yield a volume, as it merely results in a line. The inquiry emphasizes the need for clarity in defining the rotation of the curve.

PREREQUISITES
  • Understanding of calculus, specifically volume of revolution concepts
  • Familiarity with continuous functions and their properties
  • Knowledge of the disk and washer methods for volume calculation
  • Basic grasp of mathematical notation and interval notation
NEXT STEPS
  • Research the disk method for calculating volumes of revolution
  • Explore the washer method for functions with gaps or holes
  • Study examples of continuous functions and their volumes when rotated
  • Learn about the implications of discontinuities in volume calculations
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as educators looking to explain volume of revolution concepts effectively.

cbarker1
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I am thinking about how to find the volume rotate around its function.Let f be a function of x in the interval [a,b] . The function could be any curve. And the curve is rotation around itself. Would there exist a volume of the curve? And how to find the volumeThank you

CBARKER1
 
Last edited:
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Well, here are some thoughts. We need some regularity on the function, such as continuity, otherwise this can be very messy and the question won't make sense. Second, what do you mean by "rotated around itself"? If you consider a simple case, i.e. a constant function, there's no volume. You're just rotating a line.
 

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