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

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Homework Help Overview

The discussion revolves around finding the volume of the solid defined by the equation x^4+y^4+z^4=1, which falls within the subject area of multivariable calculus and geometric analysis.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various coordinate systems, including cylindrical and spherical coordinates, and express concerns about the complexity of the resulting integrals. There are attempts to clarify the setup for integration and questions about finding appropriate limits for the variables involved.

Discussion Status

The conversation reflects a mix of uncertainty and exploration, with some participants suggesting methods while others express skepticism about the feasibility of the integration. There is no explicit consensus on a single approach, but several lines of reasoning are being pursued.

Contextual Notes

Some participants mention the use of computational tools like Mathematica for integration, indicating that the problem may involve advanced functions such as error functions or elliptic integrals. There is also a reference to the difficulty of integrating by hand.

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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!
 
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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.
 
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...
 
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!
 
but there must be a way to do this... no one has any ideas?
 
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:

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