Thread: 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?

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 and see if we can find a pattern

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



So if we integrate out the y's the projection onto the x axis is

So this gives the integral



Now in three dimensions we get



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



I bet you can guess what the next one will be



Now this projection will be the unit sphere in