To find: the more wood content of the ring from two rings.
The both napkin rings are having a same quantity of wood.
The volume only depends up on the height of the napkin ring.
Hence, the both napkin rings are having same quantity of wood.
To check: The part (a) answer by computing the volume of a napkin ring using cylindrical shells.
The volume of a napkin ring is
The height of the napkin ring is h.
The radius of the napkin ring is r.
The radius of the sphere is R.
Consider the equation of napkin ring as follows:
The region lies between and .
Sketch the solid region as shown below in Figure 1.
Refer Figure 1
Calculate the volume using the method of cylindrical shell as follows.
Substitute r for a, R for b, and for in equation (2).
Differentiate both sides of the equation.
Calculate the lower limit value of u using equation (4).
Substitute r for x in equation (4).
Calculate the upper limit value of u using equation (4).
Substitute R for x in equation (4).
Apply lower and upper limits for u in equation (3).
Substitute u for and for dx in equation (3).
Integrate equation (5).
Substitute for in equation (6).
Hence, the volume of a napkin ring is .
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