5. Double integral

5. Double integral - 5. Double Integral. Now we consider...

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Unformatted text preview: 5. Double Integral. Now we consider theory of integration. As it was early we start from indefinite integrals. Partial Antiderivatives. Let as given a two-variable function f ( x, y ) on some rectangle Q = [ a, b ; c, d ] . Considering it as the function of variable x on the segment [ a, b ] we can find its antiderivative and indefinite integral: Z f ( x, y ) dx = F ( x, y ) + c ( y ) , where F ( x, y ) is called partial antiderivative of f ( x, y ) , i.e. such function that F ( x, y ) x = f ( x, y ) , the role of arbitrary constant any function c ( y ) of variable y . Similarly considering the function f ( x, y ) as the function of variable y on the segment [ c, d ] we have Z f ( x, y ) dy = G ( x, y ) + c ( x ) , G ( x, y ) y = f ( x, y ) . Definite integral with respect of x on the segment [ a, b ] is the function of y : Z b a f ( x, y ) dx = F ( x, y ) fl fl fl fl b a = F ( b, y )- F ( a, y ) = I ( y ) . Similarly Z d c f ( x, y ) dy = G ( x, y ) fl fl fl fl d c = G ( x, d )- G ( x, c ) = J ( x ) . Further, we can take the definite integral of the function I ( y ) on the segment [ c, d ] : Z d c I ( y ) dy = Z d c Z b a f ( x, y ) dx dy. 1 It is called repeated (or iterated) integral of the function f ( x, y ) on the rectangle Q = [ a, b ; c, d ] . Second repeated integral of the function f ( x, y ) on the rectangle Q = [ a, b ; c, d ] is Z b a J ( x ) dx = Z b a Z d c f ( x, y ) dy dx. Both repeated integrals are equal if the function f ( x, y ) is continuous on the rectangle Q . Examples. Z ( x 2 sin ( x 3 y )- p y- 1 + 2 x + x- 1 / 2 y 2 ) dx = =- 1 3 y cos ( x 3 y )- p y- 1 x + x 2 + 2 x 1 / 2 y 2 + g ( y ) , Z ( x 2 sin ( x 3 y )- p y- 1 + 2 x + x- 1 / 2 y 2 ) dy = =- 1 x cos ( x 3 y )- 2 3 ( y- 1) 3 / 2 + 2 xy + 2 3 x- 1 / 2 y 3 + g ( x ) , Z 1 ( x 2 y- 5 xy 4 ) dx = ( 1 3 x 3 y- 5 2 x 2 y 4 ) fl fl fl fl 1 = 1 3 y- 5 2 y 4 , Z 1 ( x 2 y- 5 xy 4 ) dy = ( 1 2 x 2 y 2- xy 5 ) fl fl fl fl 1 = 1 2 x 2- x, Z 1- 1 Z 1 ( x 2 y- 5 xy 4 ) dx dy = Z 1- 1 ( 1 3 y- 5 2 y 4 ) dy = ( 1 6 y 3- 1 2 y 5 ) fl fl fl fl 1- 1 =- 2 3 , Z 1- 1 Z 1 ( x 2 y- 5 xy 4 ) dy dx = Z 1- 1 (...
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This note was uploaded on 04/19/2011 for the course ECON 2083 taught by Professor Khan during the Spring '11 term at BC.

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5. Double integral - 5. Double Integral. Now we consider...

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