MAT 21B Lecture 9: lecture6_2s
6.2: Volume by cylindrical shells
Motivation: Recall that the volume of a solid is given by
V=Zb
a
A(x)dx
•Solids of revolution: the Disk Method:
V=Zb
a
πR(x)2dx
•Solids of revolution: the Washer Method:
V=Zb
a
A(x)dx =πZb
a
[R(x)2−r(x)2]dx
What happens for the following region rotates about the line x=L?
Trouble: May not easy (or impossible) to find x=r(y) and x=R(y).
Answer: Need a new method: SHELL Method
V≈X
k
Vk
What is Vkthen?
1
83
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Motivation: recall that the volume of a solid is given by: solids of revolution: the disk method: R(x)2dx: solids of revolution: the washer method: Trouble: may not easy (or impossible) to nd x = r(y) and x = r(y). Vk 2 ( shell radius )( shell height ) x. 2 ( shell radius )( shell height )dx a. Draw the region and sketch a line segment across it parallel to the axis of revolution. Shell radius=the distance from the axis of revolution to the segment. Shell radius = x l and height = f (x) Example: the region enclosed by the curve y = x(1 x) on [0, 1] and the x axis is revolved about the y axis to generate a solid. Example: the region enclosed by the curve y = sin2 x x to generate a solid. Find the volume of the solid. on [0, ] and the x axis is revolved about the y axis.
