Did you pick the correct one? The indefinite integral is a family of functions with a given derivative and the definite integral is the area under a curve. Incredibly, they are related to each other, as we will soon discover!
What method have we used to approximate the definite integral (starts with a "t")? This method uses a strategy developed by Archimedes, considered by many to have been the greatest applied mathematician of antiquity. His method for finding areas under curves laid the groundwork for the invention of calculus by Newton and Leibniz two thousand years later. Read this recent New York Times article by Steven Strogatz, a professor of applied mathematics at Cornell University. Professor Strogatz discusses Archimedes method of exhaustion and many of the ideas we have covered this year: Zeno's Paradox of Motion, infinity, linear approximations, and the underlying basis for calculus - the concept of a limit. Post a short comment on one "take-away" from the article.
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This article by Steven Strogatz was interesting enough to refresh my interest in Calculus. By starting with simple geometry concept of measuring circle, this professor simply derived the topic to necessity of Calculus. This article was good summary and reminder of our geometry, and how to prove the equation involving the area of circle by using the concept of infinity.
ReplyDelete"buy is it really necessary to use calculus to obtain something so basic? Yes, it is." (Steven Strogatz)
This article addresses a one of the many things that i have struggled with in calculus and physics all year. I feel that both calculus and physics take a certain amount of trust, both are based off of patterns and use mathematical impossibilities to discover the realities of how the universe works. For example the author talks about the ball getting infinitely close to the wall and feeling confused that math shows it will never touch, which i can totally relate to. it baffles me that the universe exists with mathematical perfection and yet when we look closely at the fine details none of it makes sense. But i realized after reading this article that it is because it all accounts for that nasty mind boggling number, infinity. The author mentions the affect that infinity plays in our mathematical rules and uses the example of pi to show how it is used to ensure mathematical perfection. I enjoyed reading this article because it made me feel a little more sane to know that i am not the only one baffled by this inconceivable perfection.
ReplyDeleteI thought this article was especially interesting, because it all came back the the concept of limits, the driving force behind calculus, "But when you take it to the limit...it becomes simple and beautiful, and everything becomes clear. That’s how calculus works at its best" (Strogatz). I was very interesting to me to see how Archimedes figured everything out and how much work he put into it. I think that the method he used to find the value of pi should be called the exhaustion method because it's so exhausting to do! I can't even imagine how long that took him, especially considering some of the restrictions of his time-- no calculator, very old version of pen and paper, candle light, etc. And it is amazing how he came up the idea of infinity. Nowadays, infinity is a given, and nobody really questions it. Even as children, when we learn to count we often go- 1, 2, 3, 4....97, 98, 99, 100, INFINITY! But back then, everything they did was based on tangible numbers, so the concept of infinity and limits must have been a huge stretch for their minds. The craziest part of all of this, to me, is how long people have been doing math like this, and that they were right!
ReplyDeleteLimit's, infinity, unfathomable concepts. It's funny how things that aren't really tangible are used to make things more understandable. For example, the equation for the area of a circle pi*r2, can only be explained by using the concept of infinity. I think this is very similar to calculus in general. At many times, calculus is confusing, and unfathomable. Yet despite it's difficulty to understand, it somehow is able to explain how to things that otherwise couldn't be. Now I'm not complaining about the difficulties of calculus too much. It may be difficult at times, but at least I don't have to do what Archimedes did. The method of exhaustion...phew, it was exhausting just reading reading about it.
ReplyDeleteI though this was a good article, but I think limit's are done best by Willie Nelson and Waylon Jennings. Take it to the limit one more time.
http://www.youtube.com/watch?v=B_zNNBsHf9E
This article was very eye opening for me and really re sparked my interest in calculus. The equation for the area of a circle has always been a friendly and integral part of math, just like a cell phone; you use it and trust it and get to know it very well, but at the same time, have no idea how it was created. Now that I think about it, the equation pi*r*r has always just been something that I knew to be true but was never told why. The best part about this article for me was how it proved this equation. By using the concept of an unlimited possibility of numbers and how that value set can be limited (all of this is in theory, of course), the article very simply explained the calculus of proving the area of a circle by showing that it was equivalent to a definite integral. This acted as a brilliant segue into the overall concepts of limits, infinity, definite integrals, and indefinite integrals. I cannot wait to see where we will go in class next!
ReplyDeleteHmm, this article is very worth to read. I know the second method of solving for pi. That's the one makes a lot of sense. The first method of sovling for pi is impressive. A little change, transformation, makes everything easier. I think that's what I got out from this article: Change. And I think that's what calculus is. However, If you think more about it, it is very sad. Everything is changing, no matter whether we consider it broadly or narrowly. Things are changing. They change from one formation to another. What's good about things is that they go to the limit. Things could last forever and ever. A paper, we burn it, it turns to arsh. However, nothing lost during the ignition. But our life is not infinte. It is so depressive, when everything goes, we stop. Our life is calculus with a limit.
ReplyDeleteIn the beginning of the year i was frustrated when first learning about infinite concepts in terms of Zeno's Paradox, how you in theory can get infinitely close to an object but never actually touch it. I did not understand and still cannot quite wrap my head around this theory. This article by Steven Strogatz addressed Zeno's Paradox as well as many other baffling concepts of calculus. I found the most interesting part of the article to be the proof of finding the area of the circle. By dividing up a circle into slices over and over and rearranging them so that the arc-shaped side alternates with the opposite side, the shape you are presented with approaches a rectangle. The straight side of the wedge is equal to the radius of the circle and the arcs on one side added together are equal to π*r because it is half of the circumference of the circle. In order to find the area of a rectangle you take the length and multiply it by the width, and so you are left with πr^2, which as we know is the area of the circle. Pretty crazy huh? I thought so at least. I also was fascinated by the number of applications of numerical analysis in today's world and so I googled it to find out a little more about it. Wikipedia made it a little bit easier for me to understand. "The point of numerical analysis is to analyze methods that are used to give approximate number solutions to situations where it is unlikely to find the real solution quickly, and to try and improve upon these methods so as to reduce the amount of error generated by computer calculation."(http://wiki.answers.com/Q/What_is_the_use_of_numerical_analysis_and_why_you_have_to_study) Hopefully this might help you guys too!
ReplyDeleteCool. I dug it. I think its cool that everything, in the end, is essentially a very well calculated approximation. In actuality, that new red Ferrari in my garage was designed by a collection of well thought out approximations. I'm with the Professor though, Zeno's halfway theory just freaks me out. Anyway, what I got from this article includes, I can from now on just approximate how much homework I should do, a wheel with straight sides works just as well, if not better, than a round one, and there are a whole lot of people out there a whole lot smarter than me.
ReplyDeleteThis article gives an interesting take on calculus that I've never thought of before. The author is saying how the main concept of calculus is infinity. Finding the area under a line, or the area of a circle is difficult without calculus due to the curved edges. Calculus and the concept of infinity make calculating these values easy and accurate. It is a very abstract idea since physically in the real world getting to infinity is impossible. I think this article explains calculus in a simple way that everyone can understand, and even helped me understand the concept a little more.
ReplyDeleteThis brought up a whole new way to think about math. It is logic and the figuring things out by thinking about them. It is more than simple plug and chug. The people on the top of the math field are figuring things out that are extremely important to life. It was inspirational to see the circles divided into smaller and smaller circles to turn an extremely complicated problem that is literally not solvable into a simple multiplication problem.
ReplyDeleteThe article put what could only be read by mathematicians and people who knew math language into pros which was helpful for me because I don't really speak math. It made it seem much more fun than the usual proof because it wasn't so bland, but was written so that people would be interested and continue to read.
Take it to the Limit was a very intriguing article that was written in a simple and easy to understand fashion. It seems like the more I look around, the more everything turns into limits! I am undecided as to whether or not this is a good thing. Steven Strogatz reinforced the importance of mathematics in essentially all careers and subjects.
ReplyDeleteMy favorite thing about this article was learning how they found pi times the radius squared. I also really enjoyed the fact that they found pi even before decimals came around. Just they way in which they discovered pi in the first place was very interesting and surprisingly simple. It was great how in Steven Strogatz gave some credit to Archimedes for the discovery/invention of calculus, because it seems that we only ever hear about Newton and Leibniz.
After each break away from school I somehow forget some of the concepts I learned, and I often come back with less energy than I had before. This article reminded me of the concept of limits we learned in the first semester by using the example of finding the value of pi, using “the method of exhaustion.” As I was reading, the question “is it really necessary to use calculus to obtain something so basic?” caught my attention throughout the reading. Calculus is learned after geometry, because there are some geometry basics we need to know before we learn calculus (e.g.: differentiation of trig functions). On the other hand, calculus is used to prove some basic theorems learned in geometry (e.g.: find the value of pi). This is a method that can be applied to real life problems. We don’t always need to know a specific subject in order to go through tough moments, but as we go on, we should keep in mind that there should be, a time when what we didn’t understand will make sense. All we need is patience, because “in each case, the strategy is to find a series of approximations that converge to the correct answer as a limit. And there’s no limit to where that’ll take us.”
ReplyDeleteI really enjoyed this article because of the way in which it makes the scary word "calculus" become a simple operation of limits and the area under a curve. The article is a nice way of reviewing the most basic components of Calculus, which I find easy to forget. The sentence which really struck me as being accurate and concise was helpful to my learning in calculus even now, "… as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus." (Strogatz). I have to admit that I was also taken in by the colorful pin wheels, I really do like visuals. I like reading articles like these because they take the information in the textbook and make it into more of a user friendly lesson.
ReplyDeleteThanks for the article Bru!
I really appreciate this article due in large part to the fact that it completely explains and sheds light on pi. The logic and origins as well as where the name, "the method of exhaustion" comes from. A very adequate name for the method. Pi is something that now is taken for granted for it to developed through these concepts created by archimedes 2000 years before Newton and Leibniz as the article said. He once agian returned to Zeno's concept of the the infintesimally decreasing halves, and this idea of never completing a whole, or when do we consider these portion a whole. This idea is such an abstract concept. I understand but until this failed to really appreciate an application, even within calculus. I appreciate the article very interesting.
ReplyDeleteThis article taught me a lot about circles. The fact that this concept was invented so many years ago is amazing. Archimedes is truly one of the best mathematicians to have lived. More helpful than the article were the pictures that it had. They helped because they were colorful and very understandable.
ReplyDelete