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Homework Statement
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]
Homework Equations
I use a few other known properties in the proof (see below)
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. :)
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