BobbyCottam.html

Bobby Cottam - click here for the Maple 11 Version

It seems from this that the pattern is changing based on the function being odd or even which makes sense. Because all the terms of the jacobian are being multiplied it is possible to break up the individual integrals for each variable and simply multiply them together after integrating them seperately.

All of the integrals are the same every time except for which is one degree higher, other than that additional terms keep being integrated an multiplied. Every time it is an even dimension the new term is being integrated is sin^(an even number) and every odd term is sin^(odd number).

Usinig the simplifcation

And having the limits go from 0 - π

the first term will always go to zero since sin(π) and sin(0) both equal 0. Thus the only part that is affected by higher n values is what the coefficient in front of the final integration is.

The final inegration for even n values is a term and for odd n terms is a sin(θ) term. This explains why the power of π only increases on even n values.

Now that we understand that we are indeed looking for two seperate functions we can examine them more carefully.

For the even coefficients we have This pattern appears to be the factorial of 1 2 and 3 respectively.

Thus the pattern for the volume of even n-dimensional sphere's

For the odd coefficients we have

It is easy to see that the numerator is simply successive powers of 2

The denominator's pattern can be found from looking at each individual integral to be all of the odd numbers from n - 0 multiplied together. With some investigating it turns out that it is the double factorial function of n

Thus the pattern for the volume of odd n-dimensional sphere's

As an interesting note during readings on hypersphere's there is a peak volume between n=5 and n=6 where the volume reaches a peak. An explination of this could be an interesting project they may or may not be in the calc 3 domain

"n-sphere." Wikipedia.org. 4 May 2007. 6 May 2008. <http://en.wikipedia.org/wiki/N-sphere>

"Double factorial." WolframMathworld. 6May 2008. 6 May 2008. <http://mathworld.wolfram.com/DoubleFactorial.html>