Volumes by Disks and Washers Or how much
Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway? ? Howard Lee 8 June 2000
Relief A Real Life Situation Damn, that’s a lotta toilet paper! I wonder how much is actually on that roll?
How do we get the answer? CALCULUS!!!!! (More specifically: Volumes by Integrals)
Volume by Slicing Volume = length x width x height Volume of a slice = Area of a slice x Thickness of a slice t A Total volume = (A x t)
Volume by Slicing Total volume = (A x t) But as we let the slices get infinitely thin, Volume = lim t 0 (A x t) VOLUME = A dt Recall: A = area of a slice
Rotating a Function x=f(y) Such a rotation traces out a solid shape (in this case, we get something like half an egg)
Volume by Slices } dt Thus, the area of a slice is r^2 A = r^2 Slice r
Disk Formula VOLUME = A dt But: A = r^2, so… VOLUME = r^2 dt “The Disk Formula”
Volume by Disks y axis Slice x = f(y) x r dy } thickness radius x Thus, A = x^2 but x = f(y) x axis and dt = dy, so. . . VOLUME = f(y)^2 dy
More Volumes Slice R f(x) r g(x) rotate around x axis dt 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”
Volumes by Washers Slice f(x) g(x) little r Big R R f(x) r g(x) dt dx Thus, A = (R^2 - r^2) = (f(x)^2 - g(x)^2) V = (f(x)^2 - g(x)^2) dx
The application we’ve been waiting for. . . 2 1 0. 5 f(x) g(x) 1 rotate around x axis
Toilet Paper f(x) 2 So we see that: 1 f(x) = 2, g(x) = 0. 5 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: 1 V= (2^2 - (0. 5)^2) dx = 3. 75 (1 - 0) = 0 3. 75
Other Applications? Feed me!!!!!! Just how much pasta can Pavarotti fit in that belly of his? ? or, If you’re a Britney fan, like say. . .
"Me 'n Britney 4 eva. "
Britney You can figure out just how much air that head of hers can hold! Approximate the shape of her head with a function,
The Recipe n Rotate n Slice n and Integrate
And people say that calculus is boring. . . On the next episode of 31 B. . . Volumes by Shells (aka TP Method) Or, why anything you do with volumes will involve toilet paper in one way or another n
- Slides: 19