Yahoo Answers: Answers and Comments for Cylindrical hole with radius r is drilled symmetrically through the center of a sphere with radius R.? [Mathematics]
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From Anonymous
enUS
Fri, 16 Dec 2011 00:00:55 +0000
3
Yahoo Answers: Answers and Comments for Cylindrical hole with radius r is drilled symmetrically through the center of a sphere with radius R.? [Mathematics]
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https://answers.yahoo.com/question/index?qid=20111216000055AAtgjI2
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From Anonymous: You can use the 'Washer Method' to fin...
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https://answers.yahoo.com/question/index?qid=20111216000055AAtgjI2
Fri, 16 Dec 2011 01:27:40 +0000
You can use the 'Washer Method' to find the volume.
http://mathworld.wolfram.com/images/epsgif/SphericalRing_1000.gif
(Just assume the hole goes all the way through)
(Also suppose the axes are x and y, and z is out of the screen)
It's a bit difficult to explain, but you can rotate the 'area' outside the cylinder and inside the sphere, about the y axis to get the answer.
Finding this area in terms of y:
It's the shape of a washer, outer radius (let it be r') varying with y (height), and inner radius constant as r.
The outer radius varies with y as r' = √(R²  y²)
So, the area of the 'washer' would be π ( r'²  r² ) = π (R²  r²  y²)
Integrating this with respect to y:
y varies from √(R²  r²) to +√(R²  r²)
of:
∫ π (R²  r²  y²) dy
Integrating and substituting limits, we get
Volume = (4/3) π (R²  r²) (√(R²  r²) )

From Dambarudhar: Radus of the Sphere = R
Volume of the Sphere...
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https://answers.yahoo.com/question/index?qid=20111216000055AAtgjI2
Fri, 16 Dec 2011 00:32:08 +0000
Radus of the Sphere = R
Volume of the Sphere = V = (4/3) π R³
Radius of the cylinder = r
Height of the cylinder = h
Draw the figure. From the figure it is observed = R²  r² = (h/2)²
=> h² = 4(R²  r²)
=> h = 2√(R²  r²)
Volume of the cylindrical hole = v = π r² { 2√(R²  r²) }
Volume of the remaning material = V  v = π{ (4/3) R³  2 r²√(R²  r²)}
= 2π { (2/3)R³  r²√(R²  r²) }

From rneal181: you can easily derive the formula the formula ...
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https://answers.yahoo.com/question/index?qid=20111216000055AAtgjI2
Fri, 16 Dec 2011 00:27:32 +0000
you can easily derive the formula the formula of the remaining material.
V = volume of the spherical segment  volume of the cylinder
the formula is,
V = 8pi/6[(R^2  r^2)^(3/2)]
note:
x = multiplication sign
sqrt = square root