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23-Double Integrals (General Regions)

23-Double Integrals (General Regions) - Double Integrals...

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Double Integrals: General Region I John E. Gilbert, Heather Van Ligten, and Benni Goetz As seen previously, essentially the only di ffi culty 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. It’s best to treat each region D on its own merits.
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Example 1: evaluate the integral I = D ( x + y ) dxdy when D consists of all points ( x, y ) such that 0 y 9 - x 2 , 0 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. The conditions 0 y 9 - x 2 , 0 x 3 , then show that D
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