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Double and Iterated Integrals over Rectangles

# Double and Iterated Integrals over Rectangles - Theorem...

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13.1 Double and Iterated Integrals over Rect- angles In R 2 : Motivation: Area under the curve y = f ( x ) Partition [ a, b ] into n subintervals Δ x = A k = Total Area Exact Area = Defn. Deﬁnite Integral of y = f ( x ) on [ a, b ] Z b a f ( x ) dx = lim n →∞ n X k =1 f ( x k x where Δ x is the width of each subinterval and x k is some point (ex: midpoint, left or right endpoint) in the k th subinterval 1

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In R 3 : Motivation: Volume under the surface z = f ( x, y ) and above the xy -plane 1. Partition R into n rectangular pieces 2. Construct a solid above each little rectangle V k = Total Volume Exact Volume = Defn. Deﬁnite Double Integral of y = f ( x, y ) over a rectangle R ( a x b, c y d ) where Δ A k 0 as n → ∞ and Δ A k is the area of the k th subrectangle of the xy -plane and ( x k , y k ) is some point in the k th subrectangle 2

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Unformatted text preview: Theorem. Fubini’s Theorem If f ( x, y ) is continuous on the rectangular region R : a ≤ x ≤ b , c ≤ y ≤ d , then Notes about the double integral of f ( x, y ) over a rectangle R 1. 2. Think of a double integral as the integral of an integral. Calculate from the inside out. Example 1 Z 6 Z 3 1 2 x + 3 y + 5 dydx 3 Example 2 Z 3 1 Z 6 2 x + 3 y + 5 dxdy Do. Z 2-1 Z 3 1 xy 2 dydx 4 Example 3 ZZ R xye xy 2 dA , R : 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 Example 4 Find the volume of the solid bounded by the elliptic paraboloid z = 1+( x-1) 2 +4 y 2 , the planes x = 3 and y = 2 and the coordinate planes. 5...
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Double and Iterated Integrals over Rectangles - Theorem...

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