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20e-lecture16

# 20e-lecture16 - Lecture 16 Wednesday May 6th 16.1 Improper...

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Lecture 16 - Wednesday May 6th 16.1 Improper double integrals An improper integral is an integral R R D fdA such that D is an unbounded region or f is an unbounded function. For example, if D = R 2 or D = { ( x,y ) : | x - y | ≥ 1 } , then D is unbounded. Similarly, the integral of 1 / (1 - x 2 - y 2 ) over the unit disc D = { ( x,y ) : x 2 + y 2 1 } is also an improper integral. In this section, we shall only deal with improper integrals of functions over rectangles or simple regions D , such that f has only ﬁnitely many discontinuities in D . We recall that if f : R R is continuous at all points in [ a,b ] except at some point z [ a,b ], then we may determine the integral of f on [ a,b ] using limits: Z b a f ( x ) dx = lim c z - Z c a f ( x ) dx + lim c z + Z b c f ( x ) dx when each of the integrals on the right exists. In other words, we take a ball around z and then let the radius of the ball shrink to zero in the limit. This deals with the single integral of a function which may be unbounded at a point z , and we can extend this to cases where the function has ﬁnitely many discontinuities. For double integrals, we proceed in a similar way. Suppose z is a discontinuity of a function f that is continuous at every other point of a bounded rectangle or simple region D . Let B z be an open ball of radius δ containing z . Then f is continuous on D ( δ ) = D \ B

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20e-lecture16 - Lecture 16 Wednesday May 6th 16.1 Improper...

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