LSweek29

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

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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...

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