Mean Value Theorem to calculate solids of revolution?

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SUMMARY

The discussion centers on using the Mean Value Theorem to calculate the volume of solids of revolution, specifically for the function y=e^x between x=2 and x=3. The user initially proposes using the average height derived from the Mean Value Theorem to determine the volume, but realizes this approach is flawed. The correct method involves the disk method for volume calculation and recognizing that the average value of the height does not equate to the centroid, which is essential for accurate volume computation. The user is advised to reference Pappus' theorem for further clarification.

PREREQUISITES
  • Understanding of the Mean Value Theorem in calculus
  • Familiarity with the disk method for calculating volumes of solids of revolution
  • Knowledge of the function y=e^x and its properties
  • Basic concepts of centroids and Pappus' theorem
NEXT STEPS
  • Study the disk method for calculating volumes of solids of revolution
  • Learn about Pappus' theorem and its applications in volume calculations
  • Explore the concept of centroids in relation to curves and solids
  • Practice problems involving the Mean Value Theorem and solids of revolution
USEFUL FOR

Students studying calculus independently, educators seeking to clarify concepts of volume calculation, and anyone interested in advanced applications of the Mean Value Theorem in geometry.

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Mean Value Theorem to calculate solids of revolution?

Ive been studying calculus on my own because my school doesn't offer it and i came across solids of revolution tonight. In one of the problems it says "What is the volume of the solid formed by rotating y=e^x across the x-axis between x=2 and x=3 ?" They did it using the disk method. It occurred to me however, that if you used the mean value theorem to find the average height of the curve, that would give you the average radius, so then u should just be able to use (pi)(r^2)(height) to find the volume, but height is just 3-2=1 . But it always seems to come up short. Does anybody know why?
 
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The height of the equation is e^x. I think that is the problem. Maybe not though I looked briefly.
 
The "average value" of the height of the graph is NOT the "centroid" and that's what you would need to use. Look up "Pappus' theorem" in your textbook:
 

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