Volume of a region between two spheres?

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SUMMARY

The volume of the region between two spheres, defined by the equations x² + y² + z² = 1 and X² + y² + (z-1)² = 1, can be calculated using double integrals in polar coordinates. The key step involves finding the intersection of the two spheres to establish the boundaries of the xy domain. Once the boundaries are determined, a double integral can be set up to compute the volume, taking care to identify the upper and lower surfaces correctly. This approach simplifies the problem significantly.

PREREQUISITES
  • Understanding of triple integrals
  • Familiarity with double integrals in polar coordinates
  • Knowledge of spherical coordinates
  • Ability to find intersections of geometric shapes
NEXT STEPS
  • Study the method of finding intersections of spheres
  • Learn how to set up and evaluate double integrals in polar coordinates
  • Explore applications of triple integrals in volume calculations
  • Review spherical coordinates and their applications in integration
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Students studying calculus, particularly those focusing on multivariable integration and volume calculations between geometric shapes.

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Homework Statement


Find the volume of a sphere bounded above by the sphere x^2 + y^2 + z^2 = 1 and below by the sphere X^2 + y^2 + (z-1)^2 = 1.


Homework Equations





The Attempt at a Solution


In class we have been doing double integrals with rectangular and polar, but I kinda feel like this would be a triple integral since we are going to have to consider dz dy and dx. I know sometimes we can substitute for one (for example we have done the intersection of a plane and a quadric surface such as a paraboloid) but I am really not sure how I would go about starting this.
Also, we have yet to cover integrals involving spherical coordinates yet, so I don't think that is what he is expecting us to use.

I apologize for not having a real attempt at the solution, but at the moment I am just looking for a step in the right direction so that I can hopefully get something going.

Thanks,
Ben
 
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Find the equation of the intersection of the two spheres (subtract the equations to get z). That will give you the boundary of the xy domain. Then just use a double integral with polar coordinates, taking care which is the upper and lower surface.
 
LC,
Thanks a lot. After that it was actually pretty simple. I just couldn't really think of how to get to that point, greatly appreciate the hint.
 

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