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Math Help - Computing the volume of higher dimensional polynomials? (4th, 5th, etc)

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    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?
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    Quote Originally Posted by Rumor View Post
    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 \mathbb{R}^2 and see if we can find a pattern

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

    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 [-1,1]

    So this gives the integral

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

    Now in three dimensions we get

    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

    \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

    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 \mathbb{R}^3

    \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
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