assignment5_sol

assignment5_sol - MATH1211/10-11(2)/Asol5/TNK Department of...

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Unformatted text preview: MATH1211/10-11(2)/Asol5/TNK Department of Mathematics, The University of Hong Kong MATH1211 (2010-11, Second semester) Suggested solution to Assignment 5 Note : Some solutions given below are outline only. You may need to give more details in your solution. The parts that are typed in red color are remarks for you to read; they are not a part of the solution. 1. (a) Because f is continuous on [ a,b ] and g is continuous on [ c,d ], it follows that the function h ( x,y ) = f ( x ) g ( y ) is continuous on R = [ a,b ] × [ c,d ], and by Fubini’s theorem (Theorem 2.6 on p.296 of textbook) , we have ZZ R f ( x ) g ( y ) dA = Z d c Z b a f ( x ) g ( y ) dxdy = Z d c Z b a f ( x ) g ( y ) dx dy = Z d c g ( y ) Z b a f ( x ) dx dy (because g ( y ) does not depend on x ) = Z d c g ( y ) Z b a f ( x ) dx dy = Z b a f ( x ) dx Z d c g ( y ) dy (because Z b a f ( x ) dx does not depend on y ) which is the required answer. (b) Suppose D is an elementary region of type 1 in the form of: D = { ( x,y ) | γ ( x ) ≤ y ≤ δ ( x ) , a ≤ x ≤ b } . Then ZZ D f ( x ) g ( y ) dA = Z b a Z δ ( x ) γ ( x ) f ( x ) g ( y ) dy dx =...
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assignment5_sol - MATH1211/10-11(2)/Asol5/TNK Department of...

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