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How to find the volume of the part of a sphere that protrudes from a square prism?

1

$begingroup$

I have a square prism with width $W$ and height $H$. So it's $W times W times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere of radius $R$ is also at the origin (see the picture created in Sketchup). I'm trying to find the volume of the Sphere that protrudes out of the Square prism. I can find the volume of the spherical cap that comes out the top. But I'm having trouble finding the volume around the sides, which are not just spherical caps.

I'm trying to set up an integral using spherical coordinates to find the part that extends out from the face at $y=W/2$. Then I plan to multiply this by 4 to get the total side volume. This is the integral I have now.

$$int_pi/2^3pi/2int_?^?int_W/(2sinphisintheta)^Rrho^2 sinphi drho dtheta $$

So as you can see, I don't know what limits to put on $phi$. Are there some reasonable limits I could put on $phi$ that will allow me to compute this integral? Or is there a better way to set this up? Perhaps a change of variables?

multivariable-calculus volume spherical-coordinates spheres change-of-variable

share|cite|improve this question

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think cylindrical coordinates (with rotation axis = $OX$ or $OY$) will be better.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 31 at 9:43
  

  
  
  
  

add a comment | 

1

$begingroup$

I have a square prism with width $W$ and height $H$. So it's $W times W times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere of radius $R$ is also at the origin (see the picture created in Sketchup). I'm trying to find the volume of the Sphere that protrudes out of the Square prism. I can find the volume of the spherical cap that comes out the top. But I'm having trouble finding the volume around the sides, which are not just spherical caps.

I'm trying to set up an integral using spherical coordinates to find the part that extends out from the face at $y=W/2$. Then I plan to multiply this by 4 to get the total side volume. This is the integral I have now.

$$int_pi/2^3pi/2int_?^?int_W/(2sinphisintheta)^Rrho^2 sinphi drho dtheta $$

So as you can see, I don't know what limits to put on $phi$. Are there some reasonable limits I could put on $phi$ that will allow me to compute this integral? Or is there a better way to set this up? Perhaps a change of variables?

multivariable-calculus volume spherical-coordinates spheres change-of-variable

share|cite|improve this question

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I think cylindrical coordinates (with rotation axis = $OX$ or $OY$) will be better.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 31 at 9:43
  

  
  
  
  

add a comment | 

$begingroup$

I have a square prism with width $W$ and height $H$. So it's $W times W times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere of radius $R$ is also at the origin (see the picture created in Sketchup). I'm trying to find the volume of the Sphere that protrudes out of the Square prism. I can find the volume of the spherical cap that comes out the top. But I'm having trouble finding the volume around the sides, which are not just spherical caps.

I'm trying to set up an integral using spherical coordinates to find the part that extends out from the face at $y=W/2$. Then I plan to multiply this by 4 to get the total side volume. This is the integral I have now.

$$int_pi/2^3pi/2int_?^?int_W/(2sinphisintheta)^Rrho^2 sinphi drho dtheta $$

So as you can see, I don't know what limits to put on $phi$. Are there some reasonable limits I could put on $phi$ that will allow me to compute this integral? Or is there a better way to set this up? Perhaps a change of variables?

multivariable-calculus volume spherical-coordinates spheres change-of-variable

share|cite|improve this question

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

$endgroup$

share|cite|improve this question

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

share|cite|improve this question

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748

asked Mar 31 at 7:06

Paul HurstPaul Hurst

85748