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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. Damn, that’s a lotta toilet paper! I wonder how much is actually on that roll?. A Real Life Situation. Relief. CALCULUS!!!!!. How do we get the answer?.

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volumes by disks and washers

Volumes by Disks and Washers

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

Howard Lee

8 June 2000

how do we get the answer

CALCULUS!!!!!

How do we get the answer?

(More specifically: Volumes by Integrals)

volume by slicing

Volume of a slice = Area of a slice x Thickness of a slice

t

A

Volume by Slicing

Volume = length x width x height

Total volume =  (A x t)

volume by slicing1

But as we let the slices get infinitely thin,

Volume = lim  (A x t)

t  0

Volume by Slicing

Total volume =  (A x t)

VOLUME =  A dt

Recall: A = area of a slice

rotating a function

x=f(y)

x=f(y)

Rotating a Function

Such a rotation traces out a solid shape

(in this case, we get something like half an egg)

volume by slices

Slice

Thus, the area of a slice is r^2

A = r^2

Volume by Slices

} dt

r

disk formula
Disk Formula

VOLUME =  A dt

But: A =  r^2, so…

VOLUME =   r^2 dt

“The Disk Formula”

volume by disks

radius

r

x

Volume by Disks

y axis

Slice

x = f(y)

x

dy

} thickness

x axis

Thus, A = x^2

but x = f(y)

and dt = dy, so...

VOLUME =   f(y)^2 dy

more volumes

Slice

R

r

rotate around x axis

dt

More Volumes

f(x)

g(x)

Area of a slice = (R^2-r^2)

washer formula
Washer Formula

VOLUME =  A dt

But: A =  (R^2 - r^2), so…

VOLUME =  (R^2 - r^2) dt

“The Washer Formula”

volumes by washers

Slice

Big R

little r

R

r

dt

Volumes by Washers

f(x)

f(x)

g(x)

g(x)

dx

Thus, A = (R^2 - r^2)

= (f(x)^2 - g(x)^2)

V =  (f(x)^2 - g(x)^2) dx

toilet paper

1

V =  (2^2 - (0.5)^2) dx = 3.75 (1 - 0) =

3.75 

0

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:

other applications

Just how much pasta can Pavarotti fit in that belly of his??

Other Applications?

Feed me!!!!!!

or,

If you’re a Britney fan, like say ...

britney

You can figure out just how much air that head of hers can hold!

Britney

Approximate the shape of her head with a function,

the recipe
The Recipe
  • and Integrate
  • Rotate
  • Slice
and people say that calculus is boring
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