Re: N-dimensional spheres

Quote:

Originally Posted by

**mikebreagan** Hello, all.

This is my first of I'm sure many posts here. The attached problem was the extra credit problems for my engineering calculus 3 class. They were due before the holiday break and I was unaware of them until now. Initially I was very upset that I had missed the opportunity of extra credit and then I looked at them and realized it wouldn't have mattered because I don't even know where to begin. Can someone walk me through these?

We do not help with work that counts towards your assessment, as in **extra credit**

Thread reopened as the closing date for submission is past.

CB

Re: N-dimensional spheres

For (a) we observe that:

$\displaystyle V_n(R)=\int_{B_{R,n}} dx_1 dx_2 ... dx_n$

where $\displaystyle B_{R,n}$ denotes the ball in $\displaystyle \mathbb{R}^n$ centred at $\displaystyle (0,.. ,0)$ of radius $\displaystyle R$

Then if $\displaystyle x_n=u$ we have $\displaystyle (x_1,x_2,..,x_{n-1}) \in B_{\sqrt{R^2-u^2} $, and so:

$\displaystyle V_n(R)= \int_{x_n=-R}^R \left[ \int_{B_{\sqrt{R^2-x_n^2},n-1}} dx_1 dx_2 ... dx_{n-1} \right] dx_n$

but the inner integral is just $\displaystyle V_{n-1}\left(\sqrt{R^2-x_n^2}\right)$

(b) is just (a) applied twice.

CB