What Is An Integral?
Date: 10/14/2003 at 02:05:19 From: Antony Subject: What does an integral sign thingy do? I'm in eighth grade, and curious about what an integral (S thing) does. If it is not too hard for you to explain, I would like to know. I think it has to do something with graphs and long equations, or such.
Date: 10/14/2003 at 10:15:36 From: Doctor Ian Subject: Re: What does an integral sign thingy do Hi Antony, It does have something to do with graphs. Are you familiar with functions at all? For example, if I write f(x) = x^2 that's a way of specifing the graph of a particular parabola. I can put in values for x, and get values of the function: x f(x) = x^2 --- ---------- 1 1 2 4 3 9 4 16 and so on. If I plot these on a graph, and fill in the points in between, I have the graph of a parabola: | * <- (3,f(3)) | | | | | * <- (2,f(2)) | | | * <- (1,f(1)) +----------------- 1 2 3 Does this make sense so far? Now, suppose I want to find the area between the parabola and the horizontal axis, for some part of the parabola--for example, between x=1 and x=3: | * | # | # | ## | ### | *### | #### | ##### | *###### +----------------- 1 2 3 For simple shapes like rectangles and triangles, we have formulas that we can use to find areas. But for arbitrary shapes, we use integrals to find the area. The way an integral works is by cutting the area up in to very thin rectangles, and adding up the areas of the rectangles | * ~ | 6 area = area of rectangle 1 | 6 + area of rectangle 2 | 56 + area of rectangle 3 | 456 + area of rectangle 4 | *456 + area of rectangle 5 | 3456 + area of rectangle 6 | 23456 | *123456 +----------------- When we have a finite number of rectangles, we use 'summation' notation: 6 __ ~ \ area = /__ area of rectangle i i=1 = area of rectangle 1 + ... + area of rectangle 6 Of course, when we do this, we don't get the exact area! (We put the little '~' above the '=' to remind us of that.) Some of the rectangles will stick up above the curve, and others won't quite reach it. So it will just be an approximation to the actual area. But as we make the rectangles thinner and thinner, the approximation gets closer to the true area. For example, if we cut the area into a million rectangles, we'll get a pretty good idea of the actual area! 1,000,000 __ ~ \ area = /__ area of rectangle i i=1 And as we imagine that we have an infinite number of rectangles, each with width zero, the sum 'converges', which means that after a while, we can figure out what the true value is by looking at the approximations. When that happens, we use an integral sign, to indicate that the summation is continuous rather than discrete: x=3 / area = | x^2 dx / x=1 Note that now we just have a regular '=' sign, because this is no longer an approximation. So that's the basic idea. To see it how it relates to the rest of calculus, take a look at this: A Brief Overview of Calculus http://mathforum.org/library/drmath/view/52121.html Is this what you wanted to know? Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 10/14/2003 at 20:40:11 From: Antony Subject: Thank you (What does an integral sign thingy do) Wow! I'm stunned! I really didn't think you would spend so much care with this! You guys are really an amazing website! When I asked my mom what an integral was, she started blabbing on about something I didn't get a word of. When you explained it to me, it's like WHAM, and I know it there and then! Thanks. I hope your site never fails. Sincerely, Antony.
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