LSweek29

# LSweek29 - 1 MATH1211/LSweek29/TNK/10-11(2 THE UNIVERSITY...

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Unformatted text preview: 1 MATH1211/LSweek29/TNK/ /10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 29 March 14, 2011  We solved Exercise 14 on p.308 (Section 5.2) in class. Solution : We first identify the region of integration by drawing a figure (see figure at right). Then we decompose into two sub-regions and , and write the integral as a sum of two iterated integrals to obtain the answer: ZZ D 3 y dA = ZZ D 1 3 y dA + ZZ D 2 3 y dA = Z 1 Z y 3 y dx dy + Z 3 1 Z 1 =y 2 3 y dxdy = Z 1 3 xy ¯ ¯ ¯ x = y x =0 dy + Z 3 1 3 xy ¯ ¯ ¯ x =1 =y 2 x =0 dy = Z 1 3 y 2 dy + Z 3 1 3 y dy = y 3 ¯ ¯ ¯ 1 + 3 ln y ¯ ¯ ¯ 3 1 = 1 + 3 ln 3 :  Section 5.3 . We went through Examples 1 and 2 on p.309 − 311 quickly. We note the important point on p.310, after Example 1, that for iterated integrals of non-rectangular regions, when we change the order of integration, it is not by a simple exchange of the placement of the integral signs. Instead, one must first identify the region of integration.  We partially solved Exercise 10 on p.311 (Section 5.3) in class. Partial Solution : From the given iterated integral, we identify the region D of integration as f ( x; y ) 2 R 2 j x 2 ¡ 2 · y · ¡ x; ¡ 2 · x · 1 g , which is shown...
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LSweek29 - 1 MATH1211/LSweek29/TNK/10-11(2 THE UNIVERSITY...

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