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Area/Volume of a blob

  1. Jun 9, 2005 #1
    Area/Volume of a "blob"

    I have a question. How would you find the area and volume of a figure (enclosed figure) without an equation? For example, like finding the area or volume of a lake or the area and volume of a figure with curves that can't be described with an equation. Do you use something like path integrals and multiple integrals?
     
    Last edited: Jun 9, 2005
  2. jcsd
  3. Jun 9, 2005 #2
    Both the above involve equations. You would either make simplifcations (like spherical chickens) or model the lake, probably in a piecewise fashion.
     
  4. Jun 9, 2005 #3
    If you did it in a piecewise fashion, how would you derive the equation from the arc lengths?
     
  5. Jun 9, 2005 #4
    The same way you would if it were continuous.
     
  6. Jun 9, 2005 #5
    here is one method that engineers like:

    lets say i want to know the surface area of a saddle-like shape. i don't know the equation describing the shape, perhaps because i form the shape through an industrial "stretching" process (that is nonethless well-defined).

    so, i make my saddle model, with the same thickness everywhere and using a material whose density will not vary by much.

    i then submerge the model in liquid and measure the volume of liquid displaced: this gives me the volume. i then divide by the constant thickness, and voila, we have the surface area. i can then multiply this by a scaling constant to get the surface area of the really large shape.

    given a very consistent density value, we also could have weighed the model on an accurate balance.

    for a lake: how about marking the water level immediately before it rains (to minimize the effect of evaporation), then measure the level again right after it rains. you could measure the amount of rainfall by doing the same thing with a bucket of known volume. calculate, using the bucket, the rate of rain fall per unit area. then, if you can find the area of the lake, you could use your data to calculate the lake's volume. finding the area of the lake amounts to finding the area of an irregular shape, you could use a similar empirical technique as i described above for the saddle.
     
  7. Jun 9, 2005 #6

    robphy

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    If you can sample (digitize) it [effectively, lay a grid over it],
    you could count up the number of elements (pixels) enclosed.

    It also may be possible to set-up a Monte Carlo method.
     
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