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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 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 ...
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