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24-Double Integrals II

24-Double Integrals II - Double Integrals General Region II...

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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 it’s always best to keep the integration as simple as possible, it’s 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 ) : φ ( x ) y ψ ( x ) , a x b for suitable choices of a, b and functions φ ( x ) , ψ ( x ) so that D f ( x, y ) dxdy = b a ψ ( x ) φ ( 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!
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Example 2: similarly in the case of I = D f ( x, y ) dxdy where D is shown to the right, then fixing y and integrating first with respect to x makes good
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