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Unformatted text preview: Double Integrals: General Region I John E. Gilbert, Heather Van Ligten, and Benni Goetz As seen previously, essentially the only difficulty in evaluating a double integral D f ( x, y ) dxdy when D is a rectangle [ a, b ] [ c, d ] with sides parallel to the x- and y-axes is being able to compute the single variable integrals that arise because the double integral could written as repeated single variable integrals D f ( x, y ) dxdy = d c b a f ( x, y ) dx dy = b a d c f ( x, y ) dy dx , and either choice of order of integration used, so we could always choose the more convenient one. The situation gets more complicated when D is not of the form [ a, b ] [ c, d ] , however. Its best to treat each region D on its own merits. Example 1: evaluate the integral I = D ( x + y ) dxdy when D consists of all points ( x, y ) such that y 9- x 2 , x 3 . Good First Step: always try to draw the region D . Now y 2 = 9- x 2 is a circle of radius 3 centered at the origin. Thecentered at the origin....
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This note was uploaded on 02/13/2012 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas at Austin.
- Spring '07