You can use the 'Washer Method' to find the volume.
(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)
∫ π (R² - r² - y²) dy
Integrating and substituting limits, we get
Volume = (4/3) π (R² - r²) (√(R² - r²) )
Google images/Wolfram for the image.
· 8 years ago