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Proof of the Iterated Integral Theorem
PROOF OF THE ITERATED INTEGRAL THEOREM For any region D, let B(D) be the iterated integral over D. Our plan is to prove that B has the Addition and Cylinder Properties, so that by the Uniqueness Theorem B(D) will equal the double integral. PROOF OF ADDITION PROPERTY
PROOF OF CYLINDER PROPERTY Let m be the minimum value and M the maximum value of f(x, y) on D For each fixed value of x. Integrating from a_{1} to a_{2}, But Therefore mA ≤ B(D). By a similar argument, B(D) ≤ MA. Since B has both the Addition and Cylinder Properties,


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