Why Isn't the Cylinder Slice Method Used for Surface Area of Revolution?

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SUMMARY

The discussion centers on the confusion surrounding the calculation of surface areas of revolution using integral calculus. The standard method involves using the frustum surface area formula, S=∫2πf(x)√(1+f'(x)²) dx, rather than the cylinder slice method, S'=∫2πf(x) dx. The cylindrical slice method provides the exact volume of a cylindrical shell but does not yield accurate surface area results for non-symmetric volumes. The conversation highlights the importance of understanding the geometric implications of these methods in integral calculus.

PREREQUISITES
  • Integral calculus fundamentals
  • Understanding of surface area of revolution
  • Knowledge of frustum surface area calculations
  • Familiarity with cylindrical shells in volume calculations
NEXT STEPS
  • Study the derivation of the frustum surface area formula in detail
  • Learn about the application of cylindrical shells in volume calculations
  • Explore the relationship between volume and surface area through differentiation
  • Investigate examples of non-symmetric volumes and their surface area calculations
USEFUL FOR

Students of integral calculus, mathematics educators, and anyone seeking to deepen their understanding of surface area calculations for solids of revolution.

matts0
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Hello. I just started to study integral calculus not long ago and I have some confusion when it comes to calculating the areas of surfaces of revolution using integral.
As from my testbook, when we want to calculate this kind of surface area, we often use the frustum surface area to approximate it in ds slant. Then integrate it along the defining curve( f(x) ) as the following formula states:
S=∫2πf(x)ds=∫2πf(x)√(1+f'(x)2) dx
However my problem is why we don't use the cylinder slice with dx thick to approximate it and then integrate over the specified region, as we do when calculating the volumes of solids of revolution.
And I found in this case the formula will just be:
S'=∫2πf(x) dx which is different from above.
So please someone tell me where I made a mistake. I am sorry my post is lengthy and I also wish I could explain clearer.
Thanks a lot for your time.
 
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However my problem is why we don't use the cylinder slice with dx thick to approximate it and then integrate over the specified region
,

The cylindrical slice of width dx gives the exact volume (not an approximation!) of a cylindrical shell. For a sphere, finding a volume first with cylindrical shells then taking a derivative of the volume will give the surface area of the sphere. But for other non symetric volumes that might not be so easy to do.

read more at a randomly selected thread on the subject
https://www.physicsforums.com/showthread.php?t=528358

That was pure coincidence
 
Last edited:

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