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20e-lecture14

20e-lecture14 - Lecture 14 Friday May 1st [email protected]

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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→ 0 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→ 0 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 referred to as the double integral of f over R . The Riemman Sum makes intuitive sense since f ( x I , y I

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