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Volumes Of Revolution

  1. Jul 10, 2011 #1
    As part of an assignment on Approximating Areas and Volume I am asked to derive the equation shown in the image attached.

    The question reads: "It can be shown that if y = f(x) is revolved around the x-axis to form a solid between x=a and x=b then the volume of the solid is give by the equation shown in the image.

    Some equations I have been using are basic area forumla such as
    Area (trapezium) = 1/2(a+b)xh

    I have also attempted to derive the forumula by looking at the Trapezoidal Rule and Simpson's Method and working backward to derive the formula.
     

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  2. jcsd
  3. Jul 10, 2011 #2
    As you rotate a cross section of the curve around the axis, it forms a cylinder, with radius y=f(x). and a thickness of δx.

    The volume of that cylinder is given by:
    A=[itex]\pi[/itex]y2 δx

    As the thickness of the cylinder approaches zero and you add (integrate) all the volumes of the reaaaaaly thin cylinders.
    That gives the expression:

    [itex]\int[/itex][itex]\pi[/itex]f(x)2dx

    I'm not quite sure how to put the limits in, but they are from a to b.

    I hope that helped!
     
  4. Jul 10, 2011 #3

    SammyS

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    [itex]\displaystyle \int_{a}^{b}{\pi \left(f(x)\right)^2} dx[/itex]
     
  5. Jul 10, 2011 #4
    Hi,
    Use two fonctions y=mx for a to b and y=k for b to c

    add the result and multiply by two.
     

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  6. Jul 10, 2011 #5

    SammyS

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    Actually, that should be,

    The volume of that cylinder is given by:
    A*δx = [itex]\pi[/itex]y2 δx
     
  7. Jul 11, 2011 #6
    Thanks All, But how to we actually get from A=πy2 δx to ∫πf(x)2dx
    How does the area become the volume?
     
  8. Jul 11, 2011 #7

    SammyS

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    A is the area of a circle with radius y & y = f(x). That radius goes from the x-axis, vertically up to the graph y = f(x). Multiplying times δx (delta-x) gives the volume of a very thin circular disk of thickness δx . The integral from x=a to x=b indicates that the volume of a series of such disk is summed to give the total volume of the solid of revolution.
     
  9. Jul 11, 2011 #8
    The definition of an integral is adding ([itex]\sum[/itex]) very skinny things (lim[itex]\delta[/itex] x-->0) between two points.

    If you are finding the area under a curve, you integrate between two points and are adding skinny rectangles (with almost no width i.e[itex]\delta[/itex]x), which are practically adding lines. (you just add the heights [itex]\int[/itex]ydx)

    With volumes, in this case, you are adding skinny cylinders, until the cylinder becomes practically a circle. (you just add the areas of those circles [itex]\int[/itex][itex]\pi[/itex]y2dx)

    Adding lines gives an area
    Adding areas gives a volume (just imagine adding all the areas of the pages of a book to get its volume)
     
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