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Integrals and the Fundamental Theorem (1/25/06) Today we review the concepts of definite integral, antiderivative, and the Fundamental Theorem of Calculus (which tells how the first two concepts are related). Also, we begin a discussion of “techniques of integration”, which might more accurately be called “techniques of anti-differentiation.” The first such technique, called “substitution”, tells us how to try to reverse the chain rule.

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The Definite Integral What does it mean to “integrate a function” over some part of its domain? Answer: To add up its values on that part of the domain. How can you do this?? Partial Answer: If it has only finitely many values, just add them up, giving each value appropriate “weight.” (Such a function is called a “step-function”.)

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The Integral Continued What if (as is usually true) our function has infinitely many values on the part of the domain of interest? First Answer: We can estimate its integral by using a finite number of values and giving each an appropriate “weight”. We can get better estimates by using more and more values over shorter and shorter intervals.

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The Integral in Symbols If we seek the integral of f on the interval [a,b], we can estimate it by taking n equally spaced points along the interval: a = x 0 < x 1 < …< x n-1 < x n = b. The little intervals between these are of length x = (b-a)/n. Our estimate could then be: f(x 0 ) x + f(x 1 ) x +…+ f(x n-1 ) x (“left-hand sum”) or f(x 1 ) x + f(x 2 ) x +…+ f(x n ) x (“right-hand sum”). The average of these two is called the Trapezoid Rule.

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Getting it exactly right How can we turn this estimate into the exact integral? Answer: Take the limit as n !! Another (very cool) answer: Suppose we can view our function f as the rate of change of another function F. Then the integral of f over [a,b] will simply be the total change in F over [a,b], i.e., F(b) – F(a). This is called the Fundamental Theorem of Calculus.

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The Fundamental Theorem (Part I) We are given a function f(x) on an interval [a,b] If F(x) is any antiderivative of f(x), then That is, to “add up” f ’s values from a to b, it suffices to find an antiderivative of f (not necessarily an easy thing to do!!), evaluate it at the endpoints, and subtract.

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Techniques of integration Finding derivatives involves facts and rules; it is a completely mechanical process. Finding antiderivatives is not completely mechanical. It involves some facts, a couple of rules (Sum and Difference, and Constant Multiplier), and then various techniques which may or may not work out. There are many functions (e.g., f(x) = e x^2 ) which have no known antiderivative formula.

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Antiderivative Facts x r dx = (1/(r +1)) x r +1 + C unless r = -1 (The “Reverse Power Rule” – “push up, divide”) 1/x dx = ln(x) + C a x dx = (1/ln(a)) a x + C log a (x) dx = ?????? sin(t) dt = - cos(t) + C cos(t) dt = sin(t) + C tan(t) dt = ??????

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Reversing the Chain Rule: “substitution” or “guess and check” Any ideas about x 2 (x 3 + 4) 5 dx ?? How about x e x^2 dx ? Try ln(x) / x dx But we’ve been lucky! Try sin(x 2 ) dx

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The Substitution Technique It’s called a “technique”, not a “rule”, because it may or may not work. If there is a chunk, try calling the chunk u. Compute du = (du/dx) dx Replace all parts of the original expression with things involving u (i.e., eliminate x). If you were lucky/clever, the new expression can be anti-differentiated easily.

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Assignment for Friday Read Section 5.5 of the text and go over today’s class notes, reviewing last semester’s material as needed. In Section 5.5, do Exercises 1-37 odd and 49-57 odd.

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