Ch 6 - 6 Fundamental Theorems Substitution Integration by...

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6 Fundamental Theorems, Substitution, In- tegration by Parts, and Polar Coordinates So far we have separately learnt the basics of integration and differentiation. But they are not unrelated. In fact, they are inverse operations . This is what we will try to explore in the first section, via the two fundamental theorems of Calculus. After that we will discuss the two main methods one uses for integrating somewhat complicated functions, namely integration by substitution and integration by parts . The final section will discuss integration in polar coordinates , which comes up when there is radial symmetry. 6.1 The fundamental theorems Suppose f is an integrable function on a closed interval [ a, b ]. Then we can consider the signed area function A on [ a, b ] (relative to f ) defined by the definite integral of f from a to x , i.e., (6 . 1 . 1) A ( x ) = x a f ( t ) dt. The reason for the signed area terminology is that f is not assumed to be 0, so a priori A ( x ) could be negative. It is extremely interesting to know how A ( x ) varies with x . What condi- tions does one need to put on f to make sure that A is continuous, or even differentiable? The continuity part of the question is easy to answer. Lemma 6.1 Let f, A be as above. Then A is a continuous function on [ a, b ] . Proof . Let c be any point in [ a, b ]. Then f is continuous at c iff we have lim h 0 A ( c + h ) = A ( c ) . Of course, in taking the limit, we consider all small enough h for which c + h lies in [ a, b ], and then let h go to zero. By the additivity of the integral, we 1
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have (using (6.1.1)), A ( c + h ) A ( c ) = I ( c,h ) f ( t ) dt, where I ( c, h ) denotes the closed interval between c and c + h . Clearly, I ( c, h ) is [ c, c + h ], resp. [ c + h, c ], if h is positive, resp. negative. When h goes to zero, I ( c, h ) shrinks to the point { c } , and so lim h 0 A ( c + h ) A ( c ) = 0 , which is what we needed to show. 2 Remark 6.1.2 : The general moral to remember is that, just as in real life, Integration is good and Differentiation is bad ! Indeed, as seen in this Lemma, integration makes functions better; here it takes an integrable, but not necessarily a continuous, function f , and from it obtains a continuous function. The Theorem below says that if f is in addition continuous, then its integral is even differentiable. Differentiation, on the other hand, makes functions worse. The derivative of a differentiable function f is often not differentiable (think of f ( x ) = sign( x ) x 2 at x = 0); in fact, f need not even be continuous (think of f ( x ) = x 2 sin(1 /x ) when x ̸ = 0 and = 0 when x = 0). A satisfactory answer to the question of differentiability of the integral is given by the following important result, which comes with an appropriately honorific title: Theorem 6.2 ( The first fundamental theorem of Calculus ) Let f be an integrable function on [ a, b ] , and let A be the function defined by (6.1.1).
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Ch 6 - 6 Fundamental Theorems Substitution Integration by...

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