Homework 8 - ° ∞ ° ∞ | f x y | dydx(4 nor ° ∞ °...

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APPM 2350 HOMEWORK 08 FALL 2010 Due Friday, October 22, 2010 at 4 pm under your TA’s door. 1. Consider the region R (an annulus) interior to a circle of radius 2 and exterior to a circle of radius 1. a. Using Cartesian coordinates and double integral(s), calculate the area of the annulus. Be careful when defining your limits of integration! b. Repeat the calculation above, but using double integral(s) in terms of polar coordinates. 2. On the domain x 0 , y 0, define the function f ( x, y ) such that f ( x, y ) = 0 y x 1 1 x 1 < y x 1 x < y x + 1 0 x + 1 < y . (1) a. Sketch the domain of the function f ( x, y ). Within each region, denote the value of f ( x, y ). b. Determine the values of 0 0 f ( x, y ) dydx (2) and 0 0 f ( x, y ) dxdy . (3) Are the two integrals equal? What does this mean for switching the order of integration? What theorem from your text does this example reference? c. The integrals in part (b) depend upon the order of integration because neither
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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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