When do we use 2pi vs using pi in finding the volume of a region?

Jaxon Hamilton 2022-07-15 Answered
When do we use 2 π vs using π in finding the volume of a region?
I've seen that sometimes finding the integral volume of a rotated shape in the xy-plane is multiplied by 2 π, and other times it is only multiplied by π. Can someone please tell me the difference?
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Answers (1)

Steven Bates
Answered 2022-07-16 Author has 15 answers
Step 1
Whether to use π or 2 π depends on the choice of the two common integration methods.
1) If the disk-of-circles method is used, disk area π r 2 is applied. The integration is then over a stack of disks,
V = a b π x 2 d x
Step 2
2) If the shell-of-cylinders method is used, cylinder circumference 2 π r is applied. The integration is then through cylinder walls, V = a b 2 π x y ( x ) d x
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