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April 22nd, 2015, 05:08 AM  #1 
Newbie Joined: Apr 2015 From: birmingham Posts: 1 Thanks: 0  HELP!! Maths problem has been bugging me for too long
Hello I am new to this forum and have been driven here by a problem that has been keeping me up for far too long now... If you were to take a sphere and slice through it at any point, except the center line, what would be the formula for the volume of the section of sphere you have sliced off? I have spent far too long trying to work out what seems to be a simple problem Thus far I have established the derivative of the volume of a sphere to get the gradient at any point on the remaining section but am not totally sure this is the right track to be heading down! PLEASE HELP! 
April 22nd, 2015, 07:00 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Have you taken a Calculus class? That is the technique you would want to use. Setting up a coordinate system so that the origin is at the center of the sphere, you can write its equation as $\displaystyle x^2+ y^2+ z^2= R^2$ where R is the radius of the sphere. We can take the plane slicing the sphere to be $\displaystyle z= z_0$. That plane cuts the sphere in the circle $\displaystyle x^2+ y^2+ z_0^2= R^2$ or $\displaystyle x^2+ y^2= R^2 z_0^2$, a circle of radius $\displaystyle \sqrt{(R^2 z_0^2}$. That tells us that with x going from $\displaystyle \sqrt{(R^2 z_0^2}$ to $\displaystyle \sqrt{(R^2 z_0^2}$, y goes from $\displaystyle \sqrt{R^2 z_0^2 x^2}$ to $\displaystyle \sqrt{R^2 z_0^2 x^2}$. And, for each (x, y), the height will be $\displaystyle z= \sqrt{R^2 x^2 y^2} y_0$. We can find the volume of that by $\displaystyle \int_{\sqrt{(R^2 z_0^2}}^{\sqrt{(R^2 z_0^2}} \int_{\sqrt{R^2 z_0^2 x^2}}^{\sqrt{R^2 z_0^2 x^2}}\left(\sqrt{R^2 x^2 y^2} y_0\right) dy dx$ 
April 22nd, 2015, 07:29 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2643 Math Focus: Mainly analysis and algebra 
Rather than bothering with spherical coordinates, I would look at the volume of revolution about the $x$axis between $x = a$ and $x = b$ (where $0 \lt b \lt a$) of the semicircle $f(x) = \sqrt{a^2  y^2}$. Also, take a look here. 

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