Voulme of an ice cream cone bound by a sphere

Click For Summary

Homework Help Overview

The problem involves finding the volume of an ice cream cone that is bounded by a sphere defined by the equation x²+y²+z²=1 and a surface described by z=sqrt(x²+y²-1). The context includes geometric interpretations and integration limits related to the volume calculation.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the nature of the surface defined by z=sqrt(x²+y²-1), with some identifying it as a hyperboloid rather than a cone. There are attempts to rearrange integral limits for volume calculation, and questions arise regarding the correct setup for the cone's equation and the implications of the (-1) in the square root.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the equations involved. Some guidance has been offered regarding the definition of the cone and the limits of integration, but there is no explicit consensus on how to proceed with the volume calculation.

Contextual Notes

There is confusion regarding the correct interpretation of the surface equations, particularly the presence of the (-1) in the square root, which is affecting the setup of the problem. Participants are also navigating the constraints of the problem as it relates to homework guidelines.

zimbob
Messages
4
Reaction score
0

Homework Statement


Find the volume of an ice cream cone bounded by the sphere x^2+y^2+z^2=1 and the cone z=sqrt(x^2+y^2-1)


Homework Equations


The two simultaneous equations yield x^2+y^2=1


The Attempt at a Solution



Attached
 

Attachments

Last edited:
Physics news on Phys.org
z=sqrt(x^2+y^2-1) is NOT the equation of a cone- it is a hyperboloid.

z= sqrt{x^2+ y^2) would be (the upper nappe of) a cone with vertex at the origin with sides making angle \pi/4 with the xy-plane.
 
Thanks for your response, so is it logical to re-arrange the integral limits such that it becomes:
Volume of cone =integral(limits theta= 0 to pi/4)integral(limits r=0 to 1/sqrt2)[sqrt((1-r^2)-r)dr d theta.
 
First you are going to have to define the cone part! If it is z= sqrt{x^2+ y^2}, then yes, you take, in polar coordinates, \theta= 0 to \pi/4. However, r goes from 0 to 1, not 1/\sqrt{2} because you are going up to the spherical cap.
 
The "cone" part is given as z= sqrt{x^2+ y^2-1} which I agree is not an equation for a cone but a hyperboloid as you mentioned above. What is troubling me is how to deal with the (-1) inside the sqrt.
 
Any ideas please, I am stuck.
 
Did you ever get this figured out?
I am working on the same problem with the exact same issue .. the -1.
 

Similar threads

Center of mass of spherical shell inside of cone (Apostol Problem).
  • · Replies 3 ·
Replies
3
Views
2K
Triple Integral To Find Volume Between Cylinder And Sphere
  • · Replies 2 ·
Replies
2
Views
2K
Setting the limits of an integral
  • · Replies 3 ·
Replies
3
Views
2K
Solving an Equation: Is it a Paraboloid or Cone?
  • · Replies 12 ·
Replies
12
Views
2K
How Do You Calculate the Volume Between a Cone and a Sphere?
  • · Replies 5 ·
Replies
5
Views
2K
Ice-Cream Cone problem - Volume in Spherical Coord
  • · Replies 3 ·
Replies
3
Views
2K
Find all points of intersection
  • · Replies 6 ·
Replies
6
Views
2K
Help please in understanding the limits of this integration
  • · Replies 4 ·
Replies
4
Views
2K
Describing a region using spherical coordinates
  • · Replies 5 ·
Replies
5
Views
3K
Describing a Solid Ice Cream Cone with Spherical Coordinates
  • · Replies 2 ·
Replies
2
Views
2K