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Unformatted text preview: Double Integrals: General Region II John E. Gilbert, Heather Van Ligten, and Benni Goetz In evaluating a double integral D f ( x, y ) dxdy when D is a general region in the xy-plane the choice of order of integration can depend on a number of factors. Since its always best to keep the integration as simple as possible, its always best to consider which order is preferable. Example 1: in the case of I = D f ( x, y ) dxdy where D is shown to the right, then fixing x and integrating first with respect to y makes good sense because then D = ( x, y ) : ( x ) y ( x ) , a x b for suitable choices of a, b and functions ( x ) , ( x ) so that D f ( x, y ) dxdy = b a ( x ) ( x ) f ( x, y ) dy dx . x y But if we had chosen to fix x , then the inte- gral with respect to y sometimes splits into two parts as shown in red. This would make inte- gration with respect to y more complicated! Example 2: similarly in the case of I = D f ( x, y ) dxdy where D is shown to the right, then fixing...
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