Volume of three shared cylinders

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Homework Help Overview

The problem involves calculating the volume enclosed by three intersecting cylinders defined by the equations x²+y²=1, x²+z²=1, and y²+z²=1. The context is within the subject area of multivariable calculus, specifically focusing on triple integrals and geometric interpretation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the potential use of polar coordinates and the implications of integrating with respect to z. There are questions regarding the range of z for the given equations and how it relates to the integration process.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some have suggested specific approaches, while others are questioning the assumptions made about the integration limits and the geometric properties of the shapes involved.

Contextual Notes

There appears to be some ambiguity regarding the integration limits and the choice of coordinate systems, as well as the interpretation of the cross-sections formed by the cylinders.

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Homework Statement



Calculate the volume of the space enclosed by three cylinders x2+y2=1, x2+z2=1, and y2+z2=1.

Homework Equations



Triple Integral.

The Attempt at a Solution



change to polar coordinates
 
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If x^2+z^2=1 and y^2+z^2=1 then x^2=y^2=1-z^2. The cross sections of your surface are squares integrating dz. I definitely wouldn't change to polar coordinates.
 
What is the range of z if I integrate w.r.t z?
 
physics_freak said:
What is the range of z if I integrate w.r.t z?

What's the range of z on the curve x^2+z^2=1?
 
is it (+/-) sqrt of 1-x^2?
 
physics_freak said:
is it (+/-) sqrt of 1-x^2?

Those are the values for single value of x. Suppose you consider all possible x?
 

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