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Proving Properties of Double Integrals

  1. Oct 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove the following property:
    If [itex] m <= f(x,y) <= M \hspace{2 mm} \forall (x,y) \in D,[/itex] then:
    [itex] mA(D) <= \int\int f(x,y)\,dA <= MA(D) [/itex]


    2. Relevant equations
    I use a few other known properties in the proof (see below)

    3. The attempt at a solution
    First, I should state that this problem is for Multivariable Calc. My ability to do proof's is severely lacking, and it is a skill I would like to learn (although I have yet to take a course on it, perhaps next semester). So I am hoping you all will be as critical as possible so I can learn from my mistakes. Also, I am not sure if I am using correct terminology, so please inform me if I am not. This is my attempt:
    Given that m and M are real numbers such that:
    [itex] m <= f(x,y) <=M \hspace{2 mm}\forall (x,y) \in D [/itex]
    It is known that:
    [itex] \int\int mdA <= \int\int f(x,y) \,dA <= \int\int MdA [/itex]
    Lemma:
    [itex] {if} \hspace{2 mm} f(x,y)>=g(x,y) \hspace{2 mm}\forall (x,y) \in D [/itex]
    then:
    [itex] \int\int f(x,y)\,dA >= \int\int g(x,y)\,dA [/itex]
    Lemma:
    [itex] \int\int cf(x,y)\,dA = c\int\int f(x,y)\,dA [/itex]
    then:
    [itex] m\int\int\,dA <= \int\int f(x,y)\,dA <= M\int\int\,dA [/itex]
    Lemma:
    [itex] \int\int\,dA = A(D) [/itex]
    Therefore:
    [itex] mA(D) <= \int\int f(x,y)\,dA <= MA(D) [/itex]
    Proved.
    Again, please be very critical, I want to learn how I can become better at this. :)
     
    Last edited: Oct 17, 2014
  2. jcsd
  3. Oct 17, 2014 #2

    Mark44

    Staff: Mentor

    What are m(A) and M(A)?
     
  4. Oct 17, 2014 #3
    m and M are real numbers....A(D) is the area of some closed, bounded region in the xy-plane
     
  5. Oct 18, 2014 #4

    Mark44

    Staff: Mentor

    Is the above given information? If not, how is it "known that ..."? You have cited the lemmas below, so I can see where they came from. I don't doubt that it's true, though.
    I would organize things differently. Put the lemmas at the top so that they don't interrupt the logic of your argument.
    It's given that m <= f(x, y) <= M, for all (x, y) in D,
    so ##\int\int mdA \leq \int\int f(x, y) dA \leq \int\int MdA##,
    hence ##m \int\int dA \leq \int\int f(x, y) dA \leq M\int\int dA##,
    and the conclusion follows from this.
     
  6. Oct 18, 2014 #5
    The above is not given information, thanks for pointing that out. So maybe it would work better if I stated that I am "taking the double integral over the entire region D" instead of saying "it is known"?
     
  7. Oct 18, 2014 #6
    Thanks for the advice! :)
     
  8. Oct 18, 2014 #7

    Mark44

    Staff: Mentor

    I think that's better.
     
  9. Oct 18, 2014 #8
    Awesome, thanks for the advice!
     
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