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Math Help - Multiple variable integration using polar coordinates: Quick question

  1. #1
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    Multiple variable integration using polar coordinates: Quick question

    Hello folks,

    I have a quick - and probably stupid - question regarding this particular method of integrating multiple variable functions. The volume of the region given by the formula goes until and stops at the oXY plane, right?

    Let's say I want the volume of a unit sphere. I would then have to multiply the \sqrt{x^2+y^2+C} by 2, correct?

    Thanks!
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  2. #2
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    Yes, you are correct.

    The upper half of a unit sphere is defined by the function

    <br />
\begin{aligned}<br />
z&=\sqrt{(x-a)^2+(y-b)^2}\\<br />
&=\sqrt{x^2-2ax+a^2+y^2-2by+b^2}\\<br />
&=\sqrt{x^2+y^2-2(ax+by)+a^2+b^2}.<br />
\end{aligned}<br />

    In finding the volume of the upper half of a unit sphere (say, for a=b=0), we would integrate this function in its circular domain and multiply by two.
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  3. #3
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    Quote Originally Posted by Scott H View Post
    Yes, you are correct.

    The upper half of a unit sphere is defined by the function

    <br />
\begin{aligned}<br />
z&=\sqrt{(x-a)^2+(y-b)^2}\\<br />
&=\sqrt{x^2-2ax+a^2+y^2-2by+b^2}\\<br />
&=\sqrt{x^2+y^2-2(ax+by)+a^2+b^2}.<br />
\end{aligned}<br />

    In finding the volume of the upper half of a unit sphere (say, for a=b=0), we would integrate this function in its circular domain and multiply by two.
    I believe that it would be (with the center at the origin)

    <br />
x^2+y^2+z^2=1 or for the top half

    <br />
z = \sqrt{1 - x^2 - y^2}
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  4. #4
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    Quote Originally Posted by Pierre-Alexandre View Post
    Hello folks,

    I have a quick - and probably stupid - question regarding this particular method of integrating multiple variable functions. The volume of the region given by the formula goes until and stops at the oXY plane, right?
    I don't know what you mean by that.

    Let's say I want the volume of a unit sphere. I would then have to multiply the \sqrt{x^2+y^2+C} by 2, correct?

    Thanks!
    The unit sphere is given by x^2+ y^2+ x^2= 1 or z= \pm\sqrt{1- x^2- y^2}. z= \sqrt{1-x^2-y^2} is the upper half of the sphere so, by symmetry, you could get the volume of the sphere by integrating 2 times that. I don't know where you got \sqrt{x^2+ y^2+ C} or what "C" is.
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