Would you be referring to the definition of the integral as ?
Is the epsilon-delta definition of the integral totally useless, or can it be applied in the same way as the definition of the ordinary limit? Thinking about it just twists my noodle. I've tried researching but, answers to this question aren't forthcomming. And please, if you want to tell me anything about this thing being intuitively clear, save it. I want strict proof. HELP ME!!
The definition I speak of is a little more precise:
Let be a function defined on an interval then, if for each there exists a such that
we say that is integrable on the interval and we write
This is the definition I'm talking about. I'd like to hear your thoughts. You can see that this definition parallels the definition of the ordinary limit in many ways. My beef here is that, in the the definition of the ordinary limit the number is either known, or can be found by analytic methods. But, in this definition, you can't show that a function is integrable without knowing the number which, in turn, pressuposes that you have integrated the the thing already! Do you see the circular logic here. Now I now that it can be shown that the summ is approaching a number by some other method (sequencing?), but I have not studied these methods yet. Help me a little bit here and you've definitely got a thanks coming'.
See animation first: Integration and Riemann Sums
To find the area under a curve between and , we chop the area up into space-filling rectangles. Define the width of the rectangle to be an arbitrarily small number, , and the number of rectangles used an arbitrarily large number, (notice that ). Then the area of the entire space is the same as the sum of the areas of all the little rectangles. Notice that the height of each rectangle is going to be the value of the function at the x-value that rectangle is sitting on, so the area of an individual rectangle is length X width, . Since we can get any level of accuracy here, it doesn't really matter where precisely this value of x lies.
So given and , the area , where . Now it should follow intuitively that the more rectangles you use, the more accurate the approximation will be. Therefore, we formally define the area by . Interesting trivia for you: Sir Isaac Newton used a capital "S" to denote the sum of these rectangles, which got stretched so much we invented a new symbol for it, taking the strict to the new . Likewise, we refer to the tiny widths or "differentials" by "dx" instead of .
Now, going back to our "intuition." You can see by the animation that the more rectangles we use, the less white space we get, therefore increasing our accuracy. So, let's start with an assumption: the area does exist and it is unique, call it A. Not saying we know what it is yet, just that it exists.
Now, let's say you want to find A with a precision of . That is, you seek to set up your system of rectangles such that . Our intuition tells us that any level of accuracy can be achieved, given enough rectangles. To translate this into math language, for any given , a can be found such that for all , the desired level of accuracy is achieved by this approximating sum.
The proof of this concept is quite visual, and the notation is never really consistent. Hope this helps, though.
Could you go into a little more detail on how the assumption that the value exists, and how it is that by making this assumption, we can later prove that it, in fact, does exist. Also, how would you ever be able to choose an if you don't choose the correct value of ?
Perhaps all of my questions would be answered with an example. Here's something easy.
Use the definition of the riemann integral to show that is integrable on the closed interval .
I would like this to be in the form of an proof. If that's possible.
So you're saying
Integrate first, ask questions later. If that is the only way, I'll accept it. But could someone please give me an example in terms of epsilon delta. Books are entirely devoid of this stuff.
Take a bucket outside and fill it with an arbitrary amount of water. How many liters is it? Even though you don't know exactly, certainly this figure exists, right? Since it exists, give it a name, A. Now take a measuring cup and pour the water into the cup, exactly one cup at a time, counting how many cups it holds. This simple measuring system gives you 1 cup, 2 cups, 3 cups, etc., and with each "iteration" as a mathematician would call it, the number gets closer and closer to A. Voila, when the iteration is complete, we have calculated A exactly.Could you go into a little more detail on how the assumption that the value exists, and how it is that by making this assumption, we can later prove that it, in fact, does exist.
This is not so different than what we are doing here. We look at a curve and we know that there is area under it, and that the amount of area is an unquestionably unique number. We're coming up with an iterative system of approximating this area which after a sufficient number of iterations, gives us the answer to any level of accuracy. So no, we are not assuming A exists and using this assumption to prove it does indeed exist. We know it exists, and we are attempting to calculate it to an arbitrary level of accuracy.
Here you are correct. In general, given a it is not possible to determine how many rectangles are necessary to find A with an accuracy within . This changes for different functions.Also, how would you ever be able to choose an if you don't choose the correct value of A?
Let , , , and . (Keep in mind the sum goes from to , where .)
Now recall the formula we all learned in discrete math, the story that little Gauss discovered a shortcut to adding all the numbers from 1 to 100 with this formula:
So, = =
Now, remember , so...
= = = 1
Ergo, A=1 QED.
*Note: the invention of integrals and calculus was intended as a cohesive system of legal shortcuts to allow us to have to do these kinds of calculations.
**Notice the second to last step, that . So if we want to know A within an accuracy of , well, . So in this case, for this function, .
They say a picture is worth a thousand words. Well this post is getting a little convoluted even for me.
If you are asking for an elementary epsilon delta proof of the Fundamental Theorem of Calculus (that integration is the inverse of differentiation), that is a tall order. The proof is quite lengthy and detailed.
Here is a visual argument though. Consider . According to the FTC, . Remember, is a function telling you how much area is contained under the curve on the interval . Now picture what you get by for some small : You get a narrow rectangular sliver of area of width and height . The smaller is, the more accurate: . Hence, . Look familiar?
Apologies. HallsofIvy is correct. Not all functions are integrable. But if a function is continuous on an interval, it is indeed integrable on that interval (the converse is not true). Forgive me, I have a knack for getting twisted in technicalities. A visual rule of thumb is that a function is integrable on an interval if and only if the function traps a finite amount of area under its curve inside that interval. Obviously, your example fills the bill, so we are safe in assuming A exists and we are not chasing a wild goose.You define a function to be "integrable" if and only if that A exists and then show how to find A if f is integrable- that is, how to find A if it exists.
I'm liking you more and more, Media Man. I really appreciate your help here. Generalizing the integral of 2x from 0 to x was a good show indeed. However, you spoke of a tall order regarding a proof. Well, I simply love math, and one day, I like to pursue a doctorate in the subject, so tall orders are the only orders I like to give.
But, don't worry about it know. You've done enough. Thanks.
By the way, if you are any good at Proofs. I will be posting a lot of threads on the topic. It's just one of those things that I feel that I need to master.