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Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway??

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A Real Life Situation Wow, thats a lot of 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 th) Volume of a slice = Area of a slice x Thickness of a slice A th

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Volume by Slicing Total volume = (A x th) VOLUME = A d(th) But as we let the slices get infinitely thin, Volume = lim (A x th) 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 a paraboloid) x=f(y)

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

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Disk Formula VOLUME = A d(th) VOLUME = r^2 d(th) 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 d(th) = 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) dx

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Washer Formula VOLUME = A d(th) VOL = (R^2 - r^2) d(th) 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 tummy of his?? Feed me!!!!!! or,If youre a Britney fan, like say...

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"Me 'n Britney 4 eva. (I know Mrs. Harlow didnt do this!)

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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 some people might say that calculus is boring... On the next episode of math fun... Volumes by Shells (aka TP Method)

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7.3 Volumes by Cylindrical Shells

7.3 Volumes by Cylindrical Shells

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