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Math Help - Volume of a Sphere

  1. #1
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    Volume of a Sphere

    I need to set this up as a double integral.

    x^2+y^2+z^2=1

    So the region of integration on the xy-plane should be x^2+y^2\leq 1 or,

    -1\leq x\leq 1

    -\sqrt{1-x^2}\leq y\leq \sqrt{1-x^2}

    If I take the region in the first quadrant, I end up writing

    V=\int_R\int f(x,y)dA =4\int_{0}^1\int_0^{\sqrt{1-x^2}}\sqrt{1-x^2-y^2}dydx

    The answer I have in the book gives the constant as 8, instead of 4. I'm not understanding why this is. I'm integrating over 1/4 of the region defined by  x^2+y^2\leq 1, so I'm just multiplying it by 4 to account for the rest of the region....
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  3. #3
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    I think I've figured out what's wrong. The definition of volume for two variables requires that f(x,y)> 0 so my integral only accounts for the top half above the xy-plane.
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