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LSweek30

# LSweek30 - MATH1211/LSweek30/TNK/10-11(2 THE UNIVERSITY OF...

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1 MATH1211/LSweek30/TNK/10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 30 March 21, 2011 We have the second class test. March 23, 2011 We recalled the method of change of variables for integration of single-variable functions: Theorem . Let f : [ A; B ] ! R be an integrable function over the closed interval [ A; B ] . Suppose x = x ( u ) is a differentiable function in variable u such that x maps an interval with end points a and b to the closed interval [ A; B ] in a one-to-one manner, with x ( a ) = A and x ( b ) = B . Then Z B A f ( x ) dx = Z b a f ( x ( u )) dx du du: (1) Note that on the LHS of (1), the variable of integration is x , the function f is written in terms of x , and the limits A; B of integration are limits on x ; on the RHS, the variable of integration is u , the function f is written in terms of u , and the limits a; b of integration are limits on u , but it also has an “extra” term which is dx du (which can be written as x 0 ( u ) ). “Justification” of this extra term (Note: this is not a rigorous proof; but a rigorous proof can be constructed based on this idea): We partition [ A; B ] into A = x 0 < x 1 < ¢ ¢ ¢ < x n = B , and form Riemann sums P n i =1 f ( c i x i where ¢ x i = x i ¡ x i ¡ 1 and c i 2 [ x i ¡ 1 ; x i ] . By writing the integral on the LHS of (1) as the limit of Riemann sums, we have: R B A f ( x ) dx = lim P f ( c i x i = lim P f ( x ( d i ))¢ x i , where d i are the values of u such that x ( d i ) = c i . Note that, by using differentials, ¢ x i ¼ dx du ¢ u i . Hence P f ( x ( d i ))¢ x i ¼ P f ( x ( d i ))

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