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Unformatted text preview: Lecture XV Iterated Integrals In this lecture we look at methods to compute multiple integrals, introducing iterated integrals . First, let us define the type of regions for which it can be used. These regions are called simple regions . Definition 1 A region R in E 2 is called simple if it has one of the following properties: (i) There exists an interval [ a, b ] and there exist continuous functions g 1 ( x ) and g 2 ( x ) defined on [ a, b ] such that g 1 ( x ) g 2 ( x ) for all x [ a, b ] and the region R is the set of points ( x, y ) such that a x b and g 1 ( x ) y g 2 ( y ) . A region with this property is called y-simple . (ii) There exists an interval [ c, d ] and there exist continuous functions h 1 ( y ) and h 2 ( y ) defined on [ c, d ] such that h 1 ( y ) h 2 ( y ) for all x [ c, d ] and the region R is the set of points ( x, y ) such that c y d and h 1 ( y ) x h 2 ( x ) . A region with this property is called x-simple ....
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