lec 1 - Tuesday, January 3 - Lecture 1: Integration by...

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Tuesday, January 3 Lecture 1: Integration by substitution (Refers to 6.1 in your text) After having practiced using the concepts of this lecture the student should be able to: define the differential of a function, integrate indefinite integrals by making appropriate change of variables (substitution), integrate definite integrals by appropriate change of variables. Summary of what we have learned about the notion of integration . - Our study of integration began with attempts at finding the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ]. To do this we introduced the notion of a Riemann sum. But computing areas in this way is inefficient. - The Fundamental theorem of calculus presented an alternate way to compute such numbers. This important theorem is presented into two parts. - The second part of the Fundamental theorem of calculus says that to find the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ] it suffices to find an anti-derivative of the function f ( x ), evaluating it at the limits of integration and subtracting the results. That is, if f ( x ) is continuous on [ a , b ] and F ( x ) is an anti- derivative of f ( x ) then - The first part of the Fundamental theorem of calculus says that, if f ( x ) is continuous on [ a , b ] and as x ranges over [ a , b ], then This can also be expressed by - The first part of the FTC appears to be unrelated to our initial inquiry. In these notes we used it to provide an easier proof of the second part of the FTC. But it is worth remembering it since we occasionally invoke it in particular situations. It is a more
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This note was uploaded on 03/19/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.

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lec 1 - Tuesday, January 3 - Lecture 1: Integration by...

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