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Double Integrals: Area or Volume?

  1. Apr 10, 2005 #1
    I'm reading in my Calculus book, and I see (I may see wrongly) that a double integral can describe both an Area and a Volume. Is that true? If that's true, how do I know when the Double Integral is describing an Area or a Volume? Thanks.
     
  2. jcsd
  3. Apr 10, 2005 #2
    Well, it's sort of similar to how a single integral can be used to find both area and volume (and length, as well).
     
  4. Apr 10, 2005 #3

    dextercioby

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    The typical triple integral describing a volume of a certain domain [itex] \mathcal{D}\subseteq \mathbb{R}^{3} [/itex] is

    [tex] V_{D}=\iiint_{\mathcal{D}} \ dV [/tex]

    Daniel.
     
  5. Apr 11, 2005 #4

    Galileo

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    It depends on your problem. An integral doesn't necessarily describe something geometric, but it can be used to calculate surface areas, for example:

    [tex]\iint_D \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial g}{\partial y}\right)^2}dA[/tex]
    or
    [tex]\iint_D 1 dA[/tex]

    or volumes:

    [tex]\iint_R f(x,y)dA[/tex]

    It depends on the problem.
     
  6. Apr 11, 2005 #5
    ok thanks.
     
  7. Apr 12, 2005 #6
    [tex]\iint_D \sqrt{1+\left(\frac{\partial g}{\partial x}\right)^2+\left(\frac{\partial g}{\partial y}\right)^2}\partial A[/tex]

    correct?
     
    Last edited: Apr 13, 2005
  8. Apr 12, 2005 #7

    dextercioby

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    Yeah,it was a typo by Galileo,but we usually write [itex] z=z\left(x,y\right) [/itex] when we indicate the equation of a surface in [itex] \mathbb{R}^{3} [/itex] explicitely .

    Daniel.
     
  9. Apr 12, 2005 #8
    Yeah you are correct, that was one of my problems when I was first learning surface integrals... whenever I would get stuck, I would jump to the conclusion that I could simply use that general equation to solve the problem--- but it only works in specific cases. (Or when you're scrambling on a final!)
     
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