# Thread: Multiple variable integration using polar coordinates: Quick question

1. ## 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!

2. Yes, you are correct.

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


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

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.

3. Originally Posted by Scott H
Yes, you are correct.

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


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

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)

$
x^2+y^2+z^2=1$
or for the top half

$
z = \sqrt{1 - x^2 - y^2}$

4. Originally Posted by Pierre-Alexandre
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.