Finding the Volume of x^4+y^4+z^4=1: A Challenge!

In summary, the most efficient way to find the volume of the solid x^4+y^4+z^4=1 is to use rectangular coordinates, followed by cylindrical coordinates, and finally spherical coordinates. The calculations may be complex and require the use of mathematical software, but the final answer is approximately 6.48.
  • #1
hypermonkey2
102
0

Homework Statement


What would be the most efficient way to find the volume of the solid x^4+y^4+z^4=1?


Homework Equations





The Attempt at a Solution



Cylindrical and spherical coordinates end up messy with integrals that cannot be computed by hand. I am at a loss to find something that will work in the long run! Thanks!
 
Physics news on Phys.org
  • #2
Spherical coords. You have to integrate powers of sines and cosines; look these up somewhere. Or integrate by parts. Or use e.g. cos(x)=(1/2)(e^ix + e^-ix) and expand.
 
  • #3
So i just let x=rcos(o)sin(phi) etc etc? i wouldn't need to try something like x^2=rcos(o)sin(phi)?
Because i don't see how to find limits for r in the first case...
 
  • #4
in either case, the jacobian gives me an expression that i can only integrate with mathematica using error functions or elliptical integrals... I am starting to think that this is impossible!
 
  • #5
but there must be a way to do this... no one has any ideas?
 
  • #6
Oops, sorry, it's harder than I thought. Cylindrical coords look like your best bet. Do the z integral first (easy), then the rho integral (Mathematica will do it), and finally the phi integral (ditto).

I got (pi^2 Gamma[1/4])/(3 Gamma[3/4]^3) = 6.48, which is a reasonable number (between a cube of edge length 2 and a sphere of diameter 2).

EDIT: I also got the same answer with rectangular coords, which is probably even easier.
 
Last edited:

1. What is the equation for finding the volume of x^4 + y^4 + z^4 = 1?

The equation for finding the volume of x^4 + y^4 + z^4 = 1 is a mathematical challenge that requires advanced knowledge of calculus and 3-dimensional geometry.

2. What is the significance of x^4 + y^4 + z^4 = 1 in mathematics?

This equation is significant because it represents a 3-dimensional shape known as a superellipsoid. This shape has applications in computer graphics and computer-aided design.

3. How do you approach finding the volume of x^4 + y^4 + z^4 = 1?

This equation cannot be solved using traditional methods such as integration. One approach is to use a computer program to numerically approximate the volume. Another approach is to use advanced mathematical techniques such as Monte Carlo integration.

4. Can the volume of x^4 + y^4 + z^4 = 1 be calculated exactly?

No, the volume of this shape cannot be calculated exactly due to the complexity of the equation. However, it can be approximated to a high degree of accuracy using numerical or computational methods.

5. Are there any real-world applications for x^4 + y^4 + z^4 = 1?

As mentioned earlier, this equation has applications in computer graphics and design. It can also be used to model the shape of certain objects, such as crystals, in physical sciences.

Similar threads

  • Calculus and Beyond Homework Help
Replies
21
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
459
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
571
  • Calculus and Beyond Homework Help
Replies
34
Views
1K
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
839
  • Calculus and Beyond Homework Help
Replies
4
Views
913
  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
9
Views
921
Back
Top