Volume of a solid using shells

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SUMMARY

The volume of the solid formed by rotating the region bounded by the curves y=x^(1/3) and y=x about the line y=1 is calculated using the method of shells. The correct volume is 4∏/15, as provided by the teacher, while the student's calculation of 11∏/210 indicates a misunderstanding in the setup of the problem. A sketch of the solid of revolution is essential for visualizing the volume element and ensuring accurate calculations.

PREREQUISITES
  • Understanding of solid of revolution concepts
  • Familiarity with the method of cylindrical shells
  • Knowledge of integration techniques in calculus
  • Ability to sketch curves and solids in three dimensions
NEXT STEPS
  • Review the method of cylindrical shells for volume calculations
  • Practice sketching solids of revolution for better visualization
  • Explore integration techniques for finding volumes of solids
  • Study examples of rotating regions about different axes
USEFUL FOR

Students studying calculus, particularly those focusing on volume calculations, as well as educators looking to enhance their teaching methods for solid of revolution problems.

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Homework Statement



Find the volume of the region bounded by the curve y=x^(1/3) and y=x rotated about the line y=1.

Homework Equations





The Attempt at a Solution


My teacher's solution is 4∏/15 . I got 11∏/210.

http://imageshack.us/a/img443/426/0zhs.jpg
http://imageshack.us/a/img443/426/0zhs.jpg
Where am I going wrong?
 
Last edited by a moderator:
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Feodalherren said:

Homework Statement



Find the volume of the region bounded by the curve y=x^(1/3) and y=x rotated about the line y=1.

Homework Equations





The Attempt at a Solution


My teacher's solution is 4∏/15 . I got 11∏/210.

http://imageshack.us/a/img443/426/0zhs.jpg
http://imageshack.us/a/img443/426/0zhs.jpg
Where am I going wrong?

You don't have a sketch of the solid of revolution, so I can't tell what you did. However, it looks like you didn't rotate the region around the line y = 1. I find that it's helpful to have a sketch of the solid, at least a cross-section of it, showing the typical volume element.
 
Last edited by a moderator:

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