Help w/ double integration to solve common volume of two intersecting cylinders

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Discussion Overview

The discussion revolves around finding the volume bounded by two intersecting cylinders defined by the equations x² + y² = r² and y² + z² = r². Participants explore methods for setting up double integrals to calculate this volume, considering the symmetry of the shape and the possibility of using single integration as an alternative approach.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in visualizing the integrand for a double integral and seeks suggestions for setting up the integral, noting the symmetry of the shape.
  • Another participant suggests that the integrand is 1 and proposes limits of integration for a triple integral, indicating the order of integration as dx, dy, then dz.
  • A later reply confirms the suggestion to multiply the result by 8, stating that they obtained a volume of (16/3)*r³, which they believe is correct.
  • One participant inquires about how to express the problem as a double integral instead of a triple integral.

Areas of Agreement / Disagreement

Participants generally agree on the approach of multiplying by 8 to account for symmetry, but there is no consensus on the formulation of the double integral, as one participant has specifically asked about it.

Contextual Notes

The discussion does not resolve the specifics of the double integral setup, and assumptions about the limits of integration remain unclarified. There is also a lack of consensus on the transition from triple to double integration.

Who May Find This Useful

Students studying multivariable calculus, particularly those interested in volume calculations involving intersecting geometric shapes.

Theelectricchild
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Hi I am taking MV calc and a paticular question in the double integrals chapter asks to find the volume bounded by x^2 + y^2 = r^2 and y^2 +
z^2 = r^2. I already know what the shape looks like (Steinmatic solid) and also know the answer can be achieved using single integration as well, but here I am having difficulty visualizing the integrand for a D. Integral--- the limits of integration will definitely involve constants of r. What would be your suggestion for setting up the integral? The shape is identical on all sides and symmetical--- could there be a way to solve one region and multiply the answer to get the volume, or something along those lines?

Thanks for your help.
 
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The integrand is 1. What the limits are is more interesting. do it in the order, what dx then dy then dz

x from 0 to sqrt(r^2-y^2),

y from 0 to sqrt(r^2-z^2)

z from 0 to r

multiply that answer by 8

That sound right to everyone else?
 
Multiplying by 8 makes total sense! Thanks i ended up getting (16/3)*r^3 exactly what it should be! Thanks.
 
how would u write it if it were a double integral not triple?
 

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