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Math Help - N-dimensional spheres

  1. #1
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    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?
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  2. #2
    Grand Panjandrum
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    Re: N-dimensional spheres

    Quote Originally Posted by mikebreagan View Post
    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
    Last edited by CaptainBlack; November 25th 2011 at 07:41 PM.
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  3. #3
    Grand Panjandrum
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    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
    Last edited by CaptainBlack; November 25th 2011 at 10:58 PM.
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