1. N-dimensional spheres

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?

2. Re: N-dimensional spheres

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

3. Re: N-dimensional spheres

For (a) we observe that:

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

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

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

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

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

(b) is just (a) applied twice.

CB