My Math Forum Finding Volume of a shape

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 July 20th, 2013, 10:26 AM #1 Newbie   Joined: Jul 2013 Posts: 1 Thanks: 0 Finding Volume of a shape Hi there, I am trying to find volume of a shape which surface coordinates are depicted by this: |x^n| + |y^n| + |z^n| = R^n what I have come up to find volume is: Volume = ??( R^n - |y^n| - |z^n|)^(1/n) dy dz (from 0 to R, from 0 to R) ? which should give 1/8 of total volume but it does not work. Could you please comment and explain me, why it does not work. Any help is appreciated.
 July 21st, 2013, 05:04 AM #2 Member   Joined: May 2013 From: Phnom Penh, Cambodia Posts: 77 Thanks: 0 Re: Finding Volume of a shape How can you get 1/8? The volume should be functioned to n and R.
 July 21st, 2013, 07:00 AM #3 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Finding Volume of a shape power 11110 is not saying that the volume is 1/8, he is saying that since the absolute values make the integrand symmertric about the 8 octants, it is sufficient to do this for x, y, and z all positive to get 1/8 of the entire volume. If $x^n+ y^n+ z^n= R^n$, and all of x, y, and z are positive, then, of course, $x= \sqrt[n]{R^n- y^n- z^n}$. Now, if x= 0, we have $y^n+ z^n= R^n$ and so when z= 0, $y^n= R^n$ so that y= R. That is, y goes from 0 to R and, for each y, z goes from 0 to $\sqrt[n]{R^n- y^n$. The total volume will be $8\int_0^R\int_0^{\sqrt[n]{R^n- y^n}} \sqrt[n]{R^n- y^n- z^n}dz dy$. As for actually doing that integral- you are on your own!
 July 22nd, 2013, 04:19 AM #4 Member   Joined: May 2013 From: Phnom Penh, Cambodia Posts: 77 Thanks: 0 Re: Finding Volume of a shape Sorry. I didn't understand what he was saying. Thanks, Hallsoflvy.

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