How to calculate the volume of this space

[wdlang]

in n-dimensional space, there is an area defined by

\sum_i x_i^4 <1

what is its volume?

 

[tiny-tim]

Hi wdlang!

(try using the X² and X₂ tags just above the Reply box )

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[Marin]

Hi all!

I would suggest you give it a try using the transformation formula!

The following maps could be helpful:

$$R^n\rightarrow R^n_{\geq 0}\rightarrow B^n$$, where B^n is the unit ball.

$$(x_1,...,x_n)\rightarrow (x_1^2,...,x_n^2):=(y_1,...,y_n)\rightarrow ||y||_2^2=\displaystyle\sum_{k=1}^n y_k^2=\displaystyle\sum_{k=1}^n x_k^4$$


Good luck!

 

[wdlang]

Hi all!

I would suggest you give it a try using the transformation formula!

The following maps could be helpful:

$$R^n\rightarrow R^n_{\geq 0}\rightarrow B^n$$, where B^n is the unit ball.

$$(x_1,...,x_n)\rightarrow (x_1^2,...,x_n^2):=(y_1,...,y_n)\rightarrow ||y||_2^2=\displaystyle\sum_{k=1}^n y_k^2=\displaystyle\sum_{k=1}^n x_k^4$$


Good luck!

I will try that

i am a physics student, so i am satisfied with a numerical value

so i will also try it with Monte Carlo simulation