# Computing the volume of higher dimensional polynomials? (4th, 5th, etc)

• Apr 27th 2011, 11:35 AM
Rumor
Computing the volume of higher dimensional polynomials? (4th, 5th, etc)
So, I'm working on a problem set and I'm asked to "compute the volume of the 4th dimensional unit hypersphere x^2 + y^2 + z^2 + w^2 =1" and then after that it goes on to ask me to do the same for 5th, 6th, and 7th dimensional hyperspheres.

( 5th: x^2 + y^2 + z^2 + w^2 + v^2 =1
6th: x^2 + y^2 + z^2 + w^2 + v^2 + s^2 = 1
7th: x^2 + y^2 + z^2 + w^2 + v^2 + s^2 + t^2 =1 ).

Actually computing the volume of these integrals won't be too hard, since I can use the program Maple, but it's the limits that I'm confused about. What would be the limits to the integrals? How do I figure that out?
• Apr 28th 2011, 04:45 PM
TheEmptySet
Quote:

Originally Posted by Rumor
So, I'm working on a problem set and I'm asked to "compute the volume of the 4th dimensional unit hypersphere x^2 + y^2 + z^2 + w^2 =1" and then after that it goes on to ask me to do the same for 5th, 6th, and 7th dimensional hyperspheres.

( 5th: x^2 + y^2 + z^2 + w^2 + v^2 =1
6th: x^2 + y^2 + z^2 + w^2 + v^2 + s^2 = 1
7th: x^2 + y^2 + z^2 + w^2 + v^2 + s^2 + t^2 =1 ).

Actually computing the volume of these integrals won't be too hard, since I can use the program Maple, but it's the limits that I'm confused about. What would be the limits to the integrals? How do I figure that out?

Alright lets start in $\displaystyle \mathbb{R}^2$ and see if we can find a pattern

The "volume" here will be the area of the unit circle

$\displaystyle x^2+y^2=1 \iff y=\pm\sqrt{1-x^2}$

So if we integrate out the y's the projection onto the x axis is $\displaystyle [-1,1]$

So this gives the integral

$\displaystyle \int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}dydx$

Now in three dimensions we get

$\displaystyle x^2+y^2+z^2=1 \iff z=\pm\sqrt{1-x^2+y^2}$

But now the projection of the sphere into the xy plane is the unit circle! so we get one new piece and all of the stuff from before

$\displaystyle \int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_{-\sqrt{1-x^2+y^2}}^{\sqrt{1-x^2+y^2}}dzdydx$

I bet you can guess what the next one will be

$\displaystyle x^2+y^2+z^2+w^2=1 \iff w=\pm\sqrt{1-x^2-y^2-z^2}$

Now this projection will be the unit sphere in $\displaystyle \mathbb{R}^3$

$\displaystyle \int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_{-\sqrt{1-x^2+y^2}}^{\sqrt{1-x^2+y^2}}\int_{-\sqrt{1-x^2-y^2-z^2}}^{\sqrt{1-x^2-y^2-z^2}}dwdzdydx$