Unformatted text preview: ° ∞ ° ∞  f ( x, y )  dydx (4) nor ° ∞ ° ∞  f ( x, y )  dxdy (5) exist. We say in this case that f ( x, y ) is not integrable. Thus we see that to extend Fubini’s theorem for a function on an unbounded domain, we need said function to be integrable. In contrast, show that ° ∞ ° ∞ g ( x, y ) dydx = ° ∞ ° ∞ g ( x, y ) dxdy < ∞ (6) where g ( x, y ) = 1 (1 + x 2 )(1 + y 2 ) . (7) Thus g ( x, y ) is integrable. d. From the previous problem, you might also imagine that the function h ( x, y ) = 1 1 + x 2 + y 2 (8) is integrable. Show that this is not the case by showing ° ∞ ° ∞ h ( x, y ) dydx does not exist. e. Finally, consider the function l ( x, y ) = ± 1 x ≤ 1 1 x x > 1 . (9) Show that ° ∞ ° ∞ l ( x, y ) h ( x, y ) dydx does exist....
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This note was uploaded on 02/10/2011 for the course APPM 2350 taught by Professor Adamnorris during the Fall '07 term at Colorado.
 Fall '07
 ADAMNORRIS

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