(a) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains.
(b) Express the volume in part (a) in terms of the height h of the ring. Notice that the volume depends only on h, not on r1 or r2.
To find: The volume of the given solid.
The region D lies between the sphere of radius and the cylinder of radius .
If is a polar rectangle R given by where , then, (1)
If is the function of and is the function of then,
From the given region D, let . And, r varies from to and varies from 0 to .
Substitute in the equation (1) and obtain the required volume.
Integrate the function with respect to r and by using the equation (2).
Obtain the required volume as follows
To express: The volume obtained in part (a) in terms of height of the ring.
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