hCalculus
Lesson
35
Volume
- Shells
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No different from Discs and Cross-Sections! Except now our differential volume is a shell, instead of a slab.
h
In the x-direction, if we rotate around
the y-axis, this turns into
If we rotate around the line x=4, then the length of the radius
is either x-4 or 4-x (depending on which part of the function we
are spinning around the line x=4), so a sample volume might be
Note that we are growing the cylinder
out along the radius, letting the radius
get larger and larger. However, for each
new radius, we will also have a new height!
Examples: Rotate 
on the interval 
around the
the x-axis. Then rotate about the y-axis.
Region bounded by 
rotated about the y-axis (same result as previous lesson,
but only one integral used).
Sometimes disc method is impossible to use (set up disc
method first, but we can't put x in terms of y): Region
bounded by 
and the
x-axis, rotated
about 
.