Volumes by Disks and Washers

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Volumes by Disks and Washers

• Or, how much toilet paper fits on one of those
  huge rolls, anyway??

Howard Lee 8 June 2000
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A Real Life Situation
Relief
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How do we get the answer?
(More specifically Volumes by Integrals)
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Volume by Slicing
Volume length x width x height
Total volume ? (A x ?t)
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Volume by Slicing
Total volume ? (A x ?t)
VOLUME ? A dt
Recall A area of a slice
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Rotating a Function
Such a rotation traces out a solid shape (in
this case, we get something like half an egg)
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Volume by Slices
dt
r
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Disk Formula
VOLUME ? A dt
But A ? r2, so
VOLUME ? ? r2 dt
The Disk Formula
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Volume by Disks
y axis
Slice
x f(y)
x
dy
thickness
x axis
Thus, A ?x2
but x f(y)
and dt dy, so...
VOLUME ? ? f(y)2 dy
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More Volumes
f(x)
g(x)
Area of a slice ?(R2-r2)
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Washer Formula
VOLUME ? A dt
But A ? (R2 - r2), so
VOLUME ? ?(R2 - r2) dt
The Washer Formula
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Volumes by Washers
f(x)
f(x)
g(x)
g(x)
dx
Thus, A ?(R2 - r2)
?(f(x)2 - g(x)2)
V ? ?(f(x)2 - g(x)2) dx
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The application weve been waiting for...
2
1
0.5
1
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Toilet Paper
f(x)
2
So we see that f(x) 2, g(x) 0.5
1
g(x)
0.5
1
0
V ? ?(f(x)2 - g(x)2) dx
x only goes from 0 to 1, so we use these as the
limits of integration. Now, plugging in our
values for f and g
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Other Applications?
Feed me!!!!!!
or,
If youre a Britney fan, like say ...
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"Me 'n Britney 4 eva."
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Britney
Approximate the shape of her head with a function,
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The Recipe

• Rotate

• Slice

• and Integrate

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And people say that calculus is boring...
On the next episode of 31B...
Volumes by Shells(aka TP Method)

• Or, why anything you do with volumes will involve
  toilet paper in one way or another