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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 Damn, thats a lotta toilet paper! I wonder how much is actually on that roll? Relief

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How do we get the answer? CALCULUS!!!!! (More specifically: Volumes by Integrals)

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Volume by Slicing Volume = length x width x height Total volume = (A x t) Volume of a slice = Area of a slice x Thickness of a slice A t

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Volume by Slicing Total volume = (A x t) VOLUME = A dt But as we let the slices get infinitely thin, Volume = lim (A x t) t 0 Recall: A = area of a slice

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x=f(y) Rotating a Function Such a rotation traces out a solid shape (in this case, we get something like half an egg) x=f(y)

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

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Disk Formula VOLUME = A dt VOLUME = r^2 dt But: A = r^2, so… The Disk Formula

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Volume by Disks r } thickness x axis y axisSlice radius x x dy Thus, A = x^2 x = f(y) VOLUME = f(y)^2 dy but x = f(y)and dt = dy, so...

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

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

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Volumes by Washers f(x) g(x) Slice R r dt Big R little r g(x) f(x) Thus, A = (R^2 - r^2) dx = (f(x)^2 - g(x)^2) V = (f(x)^2 - g(x)^2) dx

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

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Toilet Paper f(x) g(x) 1 2 0.5 1 So we see that: f(x) = 2, g(x) = 0.5 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: V = (2^2 - (0.5)^2) dx = 3.75 (1 - 0) = 3.75 0 1

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Other Applications? Just how much pasta can Pavarotti fit in that belly of his?? Feed me!!!!!! or,If youre a Britney fan, like say...

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"Me 'n Britney 4 eva."

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Britney You can figure out just how much air that head of hers can hold! Approximate the shape of her head with a function,

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The Recipe n and Integrate n Slice n Rotate

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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

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- Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

- Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

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