Integrals and the Fundamental Theorem (1/25/06) Today we review the concepts of definite integral, antiderivative, and the Fundamental Theorem of Calculus.
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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.
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”.)
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.
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.
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.
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.
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.
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 = ??????
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
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.
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.