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The results of integrals?

  1. Apr 17, 2015 #1
    I'm doing calclus 3 right now and I'm trying to put together the results of integrals.
    Can you correct me if I'm wrong and the one's I missed ( particularly 4 / 5 / 6). I also that the integrals can mean different things based on context. But in terms of areas and volumes atleast?

    1) ∫ dx {limits x=a to x=b} -> gives length of a line segment

    2) ∫ f(x) dx {limits x=a to x=b} -> area of region below y= f(x) >= 0 and y =0

    3) ∫ ∫ dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> area between 2 curves

    4) ∫ ∫ f(x,y) = z dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> volume bounded by the surface

    5) ∫ ∫ ∫ dx dy dz -> ???

    6) ∫ ∫ ∫ f(x,y,z) dx dy dz -> ???
     
  2. jcsd
  3. Apr 17, 2015 #2

    HallsofIvy

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    Those are possible applications of the different kinds of integrals but you should understand that the integrals themselves do not automatically give any specific application.

    ) ∫ dx {limits x=a to x=b} -> gives length of a line segment.
    This is equal to b- a which would be the length on the x-axis from a to b

    2) ∫ f(x) dx {limits x=a to x=b} -> area of region below y= f(x) >= 0 and y =0
    Assuming that y= f(x) is a graph such that f(x)> 0 for all x between a and b, then, yes, it is the area bounded by the graphs of y= f(x), y= 0, x= a, and x= b.

    3) ∫ ∫ dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> area between 2 curves
    Yes, if y= f1(x) and y= f(x2) are such that f2(x)> f1(x) (or vice-versa) for all x between a and b, this is one possible interpretation.

    4) ∫ ∫ f(x,y) = z dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> volume bounded by the surfaceY
    Yes, again with the stipulation that f(x,y) > 0 for all x, y in that area. Of course, you could also interpret f(x,y) as "surface density" so that this integral is the "mass" of the surface. More generally, f(x,y) could be a "density" of any property and the integral the total property. For example f could be temperature which is proportional to heat density and then the integral would be the heat contained in the surface.

    5) ∫ ∫ ∫ dx dy dz -> ???[/quote]
    The simplest interpretation would be the volume of the region integrated over.

    6) ∫ ∫ ∫ f(x,y,z) dx dy dz -> ???
    We can interpret f(x, y, z) as the density of some property - mass density, temperature as a "heat density", etc so that the integral gives the "total" mass or heat, etc.
     
  4. Apr 20, 2015 #3
    Thank you so much!
     
    Last edited by a moderator: Apr 20, 2015
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