Vol. common to 2 spheres of radius r, if center of each lies

[sfgiant13]

This question confuses me on all levels.

Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.

 

[arthur ohlsten]

Re: Volume of 2 Spheres

the equations of the 2 circles are
x^2+y^2=r^2
[x-r]^2+y^2=r^2

point of intersection are x=r/2 , y= r sqrt3 /2 and x=r/2 y=-r sqrt3 /2

sketch the circles , and sketch a dx dy in the intersecting area
you will integrate from [r^2-y^2]^1/2+r to [r^2-y^2]^1/2 dx
then from 0 to r sqrt3 /2 in y direction and double the answer [ or from -rsqrt3/2 to r sqrt3/2]

the answer I got was r sqrt3

Arthur

 

[skeeter]

Re: Volume of 2 Spheres

This question confuses me on all levels. Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.

wouldn't the volume common to both spheres just be double the volume of a spherical cap of one of the spheres from r/2 to r ?

\(\displaystyle V = 2\pi \int_{\frac{r}{2}}^r r^2 - x^2 \, dx\)

 

[mmm4444bot]

Re: Volume of 2 Spheres

wouldn't the volume common to both spheres just be double the volume of a spherical cap of one of the spheres from r/2 to r ?

I believe that the answer to Skeeter's question is "no" because his expression for V is AAA* correct.

~ Mark

* I acronymed "absolutely, assuredly, and altogether".

 

[arthur ohlsten]

I have been making too many errors recently.
Skeeter is correct.
I found the area common to 2 circles that overlap. I should have checked the question again befor answering.

Arthur