Volumes by Cylindrical Shells (Calculus II)

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SUMMARY

The discussion focuses on calculating the volume of a solid formed by rotating the region between the curves y = x and y = x² about the y-axis using the method of cylindrical shells. The integral setup is confirmed as correct: $$\int^1_0 (2\pi x)(x - x^2) dx$$, where 2πx represents the circumference of the shell, (x - x²) denotes the height, and dx indicates the thickness. This method effectively applies the principles of calculus to derive the volume of the specified solid.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with the method of cylindrical shells
  • Knowledge of the functions y = x and y = x²
  • Ability to perform definite integrals
NEXT STEPS
  • Study the method of cylindrical shells in detail
  • Practice calculating volumes of solids of revolution using different functions
  • Explore the comparison between cylindrical shells and disk/washer methods
  • Learn about applications of volume calculations in real-world scenarios
USEFUL FOR

Students in Calculus II, educators teaching volume calculations, and anyone interested in mastering the method of cylindrical shells for solid geometry.

shamieh
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Quick question, may seem rather dumb - but I just want to make sure of something..

Question: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2

so when I am setting up my integral am I correct in saying $$TOP - BOTTOM i.e. --> \int^1_0 (2\pi x) (x - x^2) dx$$?
 
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Re: Volumes by Cyllindrical Shells (Calculus II)

It looks good to me. The $2\pi x$ gives you the length of a shell, were you to straighten it out. The $x-x^{2}$ gives you the height of a shell, and the $dx$ gives you the thickness.
 

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