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Integration : Fubini's Theorem

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Use attached to solve the following question by integrating over an appropriate rectangle.

Assume f is class C2 Prove the following theorem by Fubini's Theorem.

Please see attachment.

For f of class C2

Left Hand side:

Right Hand side:

Use above to solve the following question by integrating over an appropriate rectangle.

Assume f is class C2 Prove the following theorem by Fubini's Theorem.
Theorem:
Let be open and suppose f: is a class C2 function.
Then for any i and j we have

(Hint: Proceed by contradiction if the mixed partials are not equal at some point, apply (a) to show we can find a rectangle on which, )
(a) Suppose is open and f: is continuous. If a and f(a) >0. there is δ>0 so that f(x) > 0 for all )

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An integral is found using Fubini's theorem.

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