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20e-lecture15

20e-lecture15 - Lecture 15 Monday May 4th [email protected]

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Key words: x -simple and y -simple regions, Fubini’s theorem, Mean value theorem Know how to interchange order of integration 15.1 Evaluating double integrals Recall that a bounded region D R 2 is y -simple if every vertical line intersects D in a line segment. A bounded region is simple if it is both x -simple and y -simple – for example the unit disc is simple. A typical example of a y -simple region is the region enclosed between two curves y = g ( x ) and y = h ( x ) for x [ a,b ], and where g ( x ) h ( x ) for all x [ a,b ]. In this case, writing dydx = dA for the diﬀerential with respect to area, we get that if f is continuous on D then Z Z D fdA = Z b a Z h ( x ) g ( x ) f ( x,y ) dydx. A similar statement holds for x -simple regions. Now this may actually not hold if f is not continuous on D . Example 1. Determine the integral of x + log y over the region D enclosed between the curves y = 1 and y = e x for 0 x 1. Solution. Observe that for 0 x 1, 1 e x . Therefore Z Z D ( x + log y ) dydx = Z 1 0 Z e x 1 ( x + log y ) dydx = Z 1 0 h x ( e x - 1) + ( y log y - y ) e x 1 i dx = Z 1 0 [( xe x - x ) + ( xe x - e x + 1)] dx = Z 1 0 [2 xe x - e x - x + 1] dx = 7 2 - e. Example 2. Determine the integral of e y /y over the region D shown below between the curves y = x and y = x for 0 x 1. 1

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20e-lecture15 - Lecture 15 Monday May 4th [email protected]

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