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Calculus II Notes 16.1

Calculus II Notes 16.1 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 16.1 16.1 Double Integrals over Rectangles We develop the double integral in a manner analogous to the definite integral for a function of one variable. Volumes and Double Integrals We used rectangles to approximate the area under a curve and developed the idea into the integral of a function of one variable. Similarly we can approximate the volume under a curve using right rectangular cylinders. Let f be continuous on the rectangle R = {( x , y ) | a x b , c y d } and let P 1 = { a = x 0 , x 1 , , x m = b } be a partition of [ a , b ] and let P 2 = { c = y 0 , y 1 , , y m = d } be a partition of [ c , d ] . Then P = P 1 × P 2 = {( x i , y j | x i P 1 , y j P 2 } is a partition of R . Note that P partitions R into m times n non-overlapping rectangles R ij = {( x i , y j ) | x i 1 x x i , y j 1 y y j } = [ x i 1 , x i ] × [ y j 1 , y j ] . Let M ij be the maximum value of f on R ij , and let m ij be the minimum value of f on R ij . Now define Δ x i = x i x i 1 and Δ y i = y i y i 1 so that the area of R ij is A ij = Δ x i

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Calculus II Notes 16.1 - Calculus II-Stewart Dr Berg Spring...

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