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

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? (More specifically: Volumes by Integrals)

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)

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

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)

Slice Thus, the area of a slice is r^2 A = r^2 Volume by Slices } dt r

Disk Formula VOLUME =  A dt But: A =  r^2, so… VOLUME =   r^2 dt “The Disk Formula”

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

Slice R r rotate around x axis dt More Volumes f(x) g(x) Area of a slice = (R^2-r^2)

Washer Formula VOLUME =  A dt But: A =  (R^2 - r^2), so… VOLUME =  (R^2 - r^2) dt “The Washer Formula”

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

f(x) g(x) rotate around x axis The application we’ve been waiting for... 2 1 0.5 1

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:

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 ...

"Me 'n Britney 4 eva."

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 • and Integrate • Rotate • Slice

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

Volumes by Disks and Washers