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Double Integrals - Volume vs. Area

  1. Jul 5, 2006 #1
    I am confused about when a double integral will give you an area, and when it will give you a volume. Since we are integrating with respect to two variables, wouldn't that always give us an area? Don't we need a third variable in order to find the volume? Thanks for the help.
  2. jcsd
  3. Jul 5, 2006 #2


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    Hello Gramma2005,

    it depends on the function you are integrating.

    Let's take a look at the function [tex]f(r)=4\pi r^2[/tex].

    Yet the following integral, (only integrating with respect to one variable!) can be interpreted as the function for the volume of a sphere depending on the radius r.

    [tex]F(r)=\int_{0}^{r} f(r') dr'[/tex]


    Last edited: Jul 5, 2006
  4. Jul 6, 2006 #3


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    A double integral will give you an area when you are using it to do that!
    A double integral is simply a calculation- you can apply calculations to many different things.

    I think that you are thinking of the specific cases
    1) Where you are given the equations of the curves bounding a region and integrate simply dA over that region. That gives the area of the region.

    2) Where you are also given some height z= f(x,y) of a surface above a region and integrate f(x,y)dA over that region. That gives the volume between the xy-plane and the surface f(x,y). It should be easy to determine whether you are integrating dA or f(x,y)dA!

    But that is only if f(x,y) really is a height. My point is that f(x,y) is simply a way of calculating things and what "things" you are calculating depends on the application. Sometimes a double integral gives pressure, sometimes mass, etc., depending on what the application is.
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