20e-lecture14

20e-lecture14 - Lecture 14 Friday May 1st [email protected] Key words double integral Riemann Sum integrable iterated integral Know definitions and

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Unformatted text preview: Lecture 14 - Friday May 1st [email protected] Key words: double integral, Riemann Sum, integrable, iterated integral Know definitions and how to evaluate iterated integrals 14.1 Definition of double integrals Recall that if f ( x ) is a function defined for a ≤ x ≤ b , then the integral of f from a to b , when it exists, is defined as the limit Z b a f ( x ) dx = lim k P k→ X I ∈ P f ( x I ) | I | where P is a partition of [ a,b ] into intervals, k P k denotes the length of the longest interval I ∈ P , and x I is an arbitrary point in I . Geometrically, if f ( x ) ≥ 0 for a ≤ x ≤ b , the integral represents the area between the curve represented by y = f ( x ) and the x-axis. We next turn to integration of functions of two variables f : R 2 → R . First, for simplicity, suppose R = [ a,b ] × [ c,d ] – this is the rectangle whose corners are ( a,c ), ( a,d ), ( b,c ) and ( b,d ). Let P be a partition of R into a grid of rectangles and let k P k be the largest side length of any of the rectangles in P . Then Z Z R fdydx = Z b a Z d c f ( x,y ) dydx = lim k P k→ X I ∈ P f ( x I ,y I ) | I | where ( x I ,y I ) is an arbitrary point in the rectangle I and | I | denotes the area of the rectangle I . The sum in the limit is referred to as a Riemann Sum and the integral is...
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture14 - Lecture 14 Friday May 1st [email protected] Key words double integral Riemann Sum integrable iterated integral Know definitions and

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