CHAPTER 07.02: TRAPEZOIDAL RULE: Multiple Segment Rule: Motivation
In this segment, we're going to talk about . . . we're going to derive the multiple segment trapezoidal rule. So let me take the example. We had f . . . we had integral going from 0.1 to 1.3, 5 x e to the power -2 x, dx, and if we were going to use one segment trapezoidal rules, let's suppose, so I'm going to show you graphically what it will look like. So your 5 x e to the power -2 x is going to look like that, as a function of x, and let's suppose if this number was 0.1, and this number is 1.3, then the single segment trapezoidal rule is just going to give you this much area here. Now, somebody might suggest that, hey, this creates a lot of true error, which is this part which is not shaded here, that what we can do is maybe we can break this up into two segments. So maybe we can do something like this. So the halfway, it's 1.4 . . . sorry, 1.2, so 0.6, so 0.7 is halfway through, and what I can do is I can draw a trapezoid like this, and I can draw another line like this, and then what I will have is I'll have a different area now under the trapezoids, so it is taking care of a lot more area. So I have this area taken care of by this first trapezoid, and this area here taken care of by this trapezoid. So you're finding out that by breaking it up into two segments, you are able to take care of more of the area which is under the curve, as opposed to just when you had only one, so this is what I get by one, and this is what I get by using two segments here. In fact, when I had one segment which was drawn, the value which I got was 0.53530, but when I have two segments, the area of this trapezoid here turns out to be 0.38175, the area of this trapezoid, the second trapezoid, turns out to be 0.40377, and the addition of the two gives you approximately 0.78550. So what you are finding out is that you are able to encompass a much larger area here, as opposed to when you had only one segment here. In fact, as I said that the exact value which you're going to get for this integral here is 0.89387, that's the exact value of the integral obtained from the integral calculus class, and this is what we got from the one segment trapezoidal rule, that's by drawing a line from point a to point b here, but if I draw the line going from point a to halfway through b, and then again all the way up to b, I get two trapezoids now. I get two trapezoids now, the area of this trapezoid turns out to be this much, the area of this trapezoid is this much, and approximately turns out to be this quantity here, and you see that it is much more . . . closer to the exact value, as opposed to the one segment one. So we can continue to do this process, we can break this up into four trapezoids, or three trapezoids, or ten trapezoids by just simply choosing different points going from 0.1 to 1.3, and be able to come up with a better and better answer, or more accurate answer for any integral we like, but that's not what we want to do by simply drawing these umpteen number of trapezoids, we want to be able to develop a formula so that somebody can program it, or somebody can simply put the numbers into the formula, and be able to get the multiple segment trapezoidal rule numbers. So let’s go ahead and see how we can derive the multiple segment trapezoidal rule. The only reason why I'm showing you this is to show you the motivation behind developing the multiple segment trapezoidal rule. |