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About this Lesson
 Type: Video Tutorial
 Length: 7:15
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 78 MB
 Posted: 06/26/2009
This lesson is part of the following series:
Calculus (279 lessons, $198.00)
Calculus: Techniques of Integration (28 lessons, $40.59)
Calculus: Numerical Integration (2 lessons, $3.96)
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
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Techniques of Integration
Numerical Integration
Example of the Trapezoidal Rule Page [1 of 2]
All right let's see how we can actually use the trapezoidal rule to actually do some numerical calculations. So what I want to do is consider the following integral. Let's look at the integral . Now first let's just sort of work this out and see what this is. So let's actually compute this really fast. Well the integral of 1 over x is the natural log of absolute value of x. And I evaluate this from 1 to 3. So when I plug in 3, I see natural log of 3 minus. And when I plug in 1, I see natural log of 1. What's the natural log of 1? Well it's the thing I have to raise e to in order to get 1, so it's 0. So in fact that's just 0. So this equals the natural log of 3.
So this is exactly the natural log of 3. But what is the natural log of 3. Suppose you actually wanted to know numerically what that was. Well how would you figure that out? Well okay you'd take out a computer or use a calculator. But suppose you were on a deserted island. In fact, suppose you were on one of those shows. Let's pretend it's like the show where you're supposed to survive. You're on an island, and all you have are sticks with flames at the end of them. In fact I've got one of those. So I'm ready now. So you're on the island and then they vote you off the island. Well suppose that they said we will not vote you off the island unless you can tell us numerically what the natural log of 3 is, at least roughly. Now here's your chance. Do you want to be voted off the island or not? And you don't have a calculator. What do you do? Well you solve this, no problem. You'll be on the problem forever. In fact, they'll leave you there, and then everyone else will leave, the camera crews, and then you're on your own. So, maybe not a great strategy. You might want to be voted off earlier. Anyway, I'll let you worry about that.
But let's see how we can actually use the trapezoid rule to approximate the natural log of 3. So let's think about what 1 over x looks like. So the graph of 1 over x, well we know what that looks like. It's just sort of our favorite hyperbola. I'll draw the relevant wing here. And we're looking at this from 1 to 3. So we're looking at this region right here. And so this integral, of course, represents the area between those two points under the curve. So we're looking at this. Now let's use the trapezoidal rule to figure out an approximation for this area if we were to divide this up into four equally spaced, equally placed trapezoids. So I want to take this, cut this into four pieces. The bases are equal. So let's use the trapezoid color. And I'm going to put in four trapezoids, all of which have the same base length. I'm trying to put this in here. You can see that now what I'm going to do is find the area of each of these four trapezoids and see what that equals. Okay, what would that be?
Well first of all let's find these points here. So I'm cutting this up into four equal pieces. So this point is right in between 1 and 2, so this is going to be 3 over 2. And between 2 and 3, this is 5 over 2. So now I want to figure out all of the areas of all these trapezoids. Now what is the formula? Well the formula is just literally to figure out the area. So what's the base of each? Well the base of each we see is just going to be . So I have a . So the area is approximately equal to . But don't forget there's that extra half, because when I actually compute the area of a trapezoid, I have to take the average value of the two extreme sides. So in fact, there's going to be another . I pull that right out in front. And then what do I see? Well I just see the value of the function at each of these points. However here I only count that value once, whereas here I have to count it twice, because I first find the average with respect to that trapezoid. And then use that side to find the average with respect to that trapezoid. So all of these interior ones are going to be doubled. And only the endpoint ones will be just there all on their own.
So let's see. The function of course is the function . So if I plug in 1 I get 1 plus. Now if I plug in into the thing, that just inverts it, and so I see , but I have to multiply that by 2 since I have two of them. So I see that. And then what do I add? Well then I add the next person. So I have to invert 2 and I see . But I've got two of them, so I have 2 x . And then I have to add to that. I invert this, which would be . By the way, why am I inverting? Because I take this thing and plug back into the function. And this is the 1 over function. So in fact, this just splits the top and the bottom. So I see 2 divided by . And then finally I just have this, which is a . I only count that once, because remember that only touches one of the trapezoids, whereas all these interior ones touch two adjacent ones.
And so that's the answer. And what is that? Well that's just . And that number, of course, even on a desert island, even in the sand, you can actually compute. And if you compute that, you'll get 67 divided by 60. So your approximation for your friends on the island would be, well you'd say, "I think the natural log of 3 is approximately 67 over 60.
Now what is that numerically? Well let's see. So now we'll actually check and see. Of course you don't have the calculator on the island with you, I know. But let's just see how we made out. So let's first compute the natural log of 3. And you can see it there. It's 1.0986 stuff. So okay, that's what it is. It's a little bit bigger than 1. Now let's take a look at what our guess is, 67 divided by 60. And what do we see there? Here we see 1.1166666. So that's incredibly close when you consider all we did was just put in four trapezoids. Four is absolutely nothing really. Imagine putting in 8 or 10 or 100 trapezoids. You'd get even more accuracy. But look what kind of accuracy we get just putting in four trapezoids. So you would definitely be voted to stay on the island because of your great utility. And thus we see another real world application of calculus. I'll see you at the next lecture.
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