Fill in the details of the following argument to prove

Chapter 12, Problem 12.27

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Fill in the details of the following argument to prove Greens theorem under special conditions. Assume that D can be described in two ways. First, D consists of all (x, y) with q(x) y p(x), for a x b. This means that D has an upper boundary (graph of y = p(x)) and a lower boundary (y = q(x)) for a x b. Also assume that D consists of all (x, y) with (y) x (y), with c y d. In this description, the graph of x = (y) is a left boundary of D, and the graph of x = (y) is a right boundary. Using the first description of D, show that C g(x, y) dy = d c g((y), y) dy + c d g((y), y) dy and D g x d A = d c (y) (y) g x d A = c c (g((y), y) g((y), y)) dy. Thus, conclude that C g(x, y) dy = D g x d A. Now use the other description of D to show that C f (x, y) dx = D f y d A. 12.3 An Exte