Calculus 7.3 Day 2

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Volumes using Disks and Washers
7.2
Limerick Nuclear Generating Station, Pottstown,
Pennsylvania
2
Suppose I start with this curve.
What happens if we revolve this curve around the
x-axis?
How would we find the volume of this figure?
3
Think of cutting it into flat cylinders (or
disks)
In this case
r the y value of the function
thickness a small change in x dx
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If we add the volumes, we get
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How do we know this works? Lets try it with
something more familiar
What shape do we get revolving this region
around the x-axis?
(r, r)
A cone.
Using calculus, the volume would be
(r, 0)
This is the pre-calc formula for volume of a cone!
In pre-calc, volume formulas are used when the
radius is constant. In calculus, we are working
with functions where the radius is constantly
changing, which is why integrals are necessary.
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(This works for any horizontal axis of rotation.)
If the shape is rotated about the y-axis (ie. the
disks are perpendicular to the y-axis), then the
formula becomes
(This works for any vertical axis of rotation.)
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Follow these steps any time you do a disk or
washer problem
1) ALWAYS draw the picture of the graph(s).
2) Draw a representative rectangle to the axis of
rotation. Is there a hole? Decide if its a disk
or washer.
3) Set up the integral and solve.
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Ex. 2
We use a disk perpendicular to the y-axis so the
thickness is dy.
volume of disk
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Ex. 3 Find the volume of the solid formed by
revolving the region bounded by
and about the line
First, note that the two graphs intersect at x
-1 and x 1.
Just like when we did area, to get the radius, we
need to subtract f(x) g(x)
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Ex. 4
The region bounded by and
is revolved about the y-axis. Find the volume.
If we use a horizontal slice
The disk now has a hole in it, making it a
washer.
outer radius
inner radius
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This application of the method of slicing is
called the washer method. The shape of the slice
is a circle with a hole in it, so we subtract the
area of the inner circle from the area of the
outer circle.
(Again, if its a vertical axis of revolution,
wed integrate with respect to y instead of x.)
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Ex. 5
If the same region is rotated about the line x2
The outer radius is
The inner radius is
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Ex. 6
We can use the washer method if we split it into
two parts
cylinder
inner radius
outer radius
thickness of slice