How Can I Visualize the Intersection of a Cone and a Sphere?

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SUMMARY

The discussion focuses on visualizing the intersection of a cone defined by the equation $z=\sqrt{x^2+y^2}$ and a sphere represented by $x^2+y^2+z^2=2$. Participants suggest using cylindrical coordinates to set up the integral for volume calculation. However, challenges arise in graphing both regions simultaneously using Wolfram Alpha. The conversation highlights the need for software that can effectively visualize and shade the intersecting volume.

PREREQUISITES
  • Cylindrical coordinates in calculus
  • Understanding of triple integrals
  • Basic knowledge of conic sections and spheres
  • Familiarity with graphing software like Wolfram Alpha
NEXT STEPS
  • Explore graphing software alternatives for 3D visualization, such as GeoGebra or MATLAB
  • Learn about using spherical coordinates for volume integration
  • Investigate techniques for visualizing intersections of 3D shapes
  • Review resources on triple integrals in spherical coordinates
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and 3D geometry, as well as software developers creating visualization tools for mathematical concepts.

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Find the volume of the solid that is enclosed by the cone $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=2$.

The integral is not difficult to set up using cylindrical coordinates, but I'm trying to get a better visualization of the volume I'm actually integrating. I can't seem to get Wolfram Alpha to graph both regions at the same time. Is there any way I can do that, or even better, have it to shade in that region?

(I have been able to graph it by hand, but it is a terrible sketch :( )
 
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