asn4 - f and g on the interval [ a,b ] . Using multiple...

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MAT237Y – Multivariable Calculus Summer 2009 Assignment # 4 Questions are worth 10 marks each. Due Tuesday, July 7th, at 6:10pm sharp. 1. Let f be defined by f ( x,y ) = ( 1 if x = 1 3 and y Q 0 otherwise and take Q to be the set [0 , 1] × [0 , 1] in R 2 . Prove that RR Q fdA exists, but that R 1 0 f ( 1 3 ,y ) dy does not exist. 2. Evaluate (a) R 1 0 R 1 y sin( x 3 + 1) dxdy. (b) RR D p 1 - x 2 - y 2 dA, where D is the disc of radius 1 in R 2 . 3. Integration by Parts: Let F and G be antiderivatives for
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Unformatted text preview: f and g on the interval [ a,b ] . Using multiple integration, show that Z b a f ( x ) G ( x ) dx = F ( b ) G ( b )-F ( a ) G ( a )-Z b a g ( y ) F ( y ) dy. (Hint: consider RR T h ( x,y ) dA for a suitable function h, where T is the triangle with vertices ( a,a ) , ( b,a ) , and ( b,b ) . ) No late assignments. 1...
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