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lect116_27rev_f11

# lect116_27rev_f11 - Tuesday November 15 Lecture 27...

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Tuesday, November 15 Lecture 27 : Integration by substitution (Refers to Section 6.1 your text) Students who have mastered the content of this lecture know : About differentials, integration by change of variable (equivalently, by substitution). Students who have practiced the techniques presented in this lecture will be able to : Compute integrals by substitution correctly. Summary of what 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

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- The first part of the FTC appears to be unrelated to our initial inquiry. In these notes
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lect116_27rev_f11 - Tuesday November 15 Lecture 27...

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