# sn10_1 - is a rectangle that contains Ω and F x y = f x y...

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Math 2433 - Week 10 Notes Popper010 1. Compute the lower Riemann sum for the given function over the interval with respect to the partition a) b) c) d) e) 2. Sketch the region bounded by the graphs of x + y 2 + 2 = 0and x + y = -4 and find its area. a) b) c) d) e)

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16.2 The Double Integral over a Rectangle Definition: Let f = f ( x, y ) be continuous on the rectangle R : a < x < b, c < y < d . Let P be a partition of R and let m ij and M ij be the minimum and maximum values of f on the i, j sub- rectangle R ij . Then ( i ) Lower sum: L f ( P ) = 1 1 n m ij i j i j m x y = = Δ Δ ∑∑ ( ii ) Upper sum: U f (P) = 1 1 n m ij i j i j M x y = = Δ Δ ∑∑ Upper and Lower sums over a rectangle: Example: Find L f ( P ) and U f ( P ) on 2 ( , ) 2 ,1 3, 0 4 f x y x y x y = - ≤ ≤ ≤ ≤ P 1 = {1, 2, 5/2, 3} and P 2 = {0,1/2, 2, 4}
The double integral of f over R is the unique number I that satisfies L f ( P ) < I < U f ( P ) for all partitions P . Notation: ( , ) I f x y dxdy = ∫∫ Let be an arbitrary closed bounded region in the plane. Then ( , ) ( , ) f x y dxdy F x y dxdy Ω = ∫∫ ∫∫ where R

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Unformatted text preview: is a rectangle that contains Ω , and F ( x, y ) = f ( x, y ) on Ω and F ( x, y ) = 0 on R- Ω . Poppper010 3. Let f be the function defined below on the given region and let P be the partition P = P 1 x P 2 . Find U f ( P ). a) b) c) d) e) 16.3 Repeated Integrals If the region Ω is given by a < x < b , 1 2 ( ) ( ) x y x φ ≤ ≤ (this is called a Type I region), then 2 1 ( ) ( ) ( , ) ( , ) x b a x f x y dxdy f x y dydx Ω = ∫∫ ∫ ∫ If the region Ω is given by c < y < d , 1 2 ( ) ( ) y x y ψ ≤ ≤ (this is called a Type II region), then 2 1 ( ) ( ) ( , ) ( , ) x d c x f x y dxdy f x y dxdy Ω = ∫∫ ∫ ∫ Application formulas: Examples: 1. 2. 3. Evaluate: a. b. 4. 5. Popper010 4. Evaluate the integral taking Ω : 0 < x < 1, 0 < y < 4. a) b) c) d) e) 5. Evaluate the integral taking Ω : 0 < y < 1, y 2 < x < y . a) b) c) d) e) 6. C...
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sn10_1 - is a rectangle that contains Ω and F x y = f x y...

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