Presentation on theme: "Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway?? Howard Lee 8 June 2000."— Presentation transcript:
Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway?? Howard Lee 8 June 2000
A Real Life Situation Damn, thats a lotta toilet paper! I wonder how much is actually on that roll? Relief
How do we get the answer? CALCULUS!!!!! (More specifically: Volumes by Integrals)
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
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
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)
Volume by Slices Slice r } dt Thus, the area of a slice is r^2 A = r^2
Disk Formula VOLUME = A dt VOLUME = r^2 dt But: A = r^2, so… The Disk Formula
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...
More Volumes f(x) g(x) rotate around x axis Slice R r Area of a slice = (R^2-r^2) dt
Washer Formula VOLUME = A dt VOLUME = (R^2 - r^2) dt But: A = (R^2 - r^2), so… The Washer Formula
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
2 The application weve been waiting for... 1 rotate around x axis 1 0.5 f(x) g(x)
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
Other Applications? Just how much pasta can Pavarotti fit in that belly of his?? Feed me!!!!!! or,If youre a Britney fan, like say...