***WARNING. THIS POST CONTAINS EXAMPLES OF PUERILE HUMOUR***
It is a sorry fact that most equations cannot be solved using neat algebra. It's probably true that most of the ones you saw at school turned out nicely but this is because you were sheltered from the harsh and uncompromising elements of the mathematical universe by your kindly teacher.
If you studied maths in the sixth-form then you would also have discovered how to find the area under a curve by integrating its corresponding function. Similarly, however, most functions can't be neatly integrated; you tend just to come across the friendly ones in school. There are, however, tried and tested methods for finding approximate areas under curves.
One such method is known as the Trapezium Rule (or, in the US, the Trapezoidal Rule) and involves calculating an estimate for the area under a curve by drawing a series of straight lines between points lying on it. This Wiki-illustration demonstrates the process:
The trapezium, while a superb method due to ease of area calculation, is far from the only shape it is possible to use. In a moment of procrastination you might, therefore, imagine my piqued interest when I clicked on a weblink labelled, "uncommon methods of numerical integration". Looking forward to a juicy academic article I was first shocked and then paralysed with laughter when the computer screen returned this:
After pulling myself together and printing it off to leave in a colleague's pigeon hole, I got round to asking the all important question: Does it work?
This post aims to put some meat on the bare bone of the above conjecture.
Firstly, the diagram above is a little ambiguous regarding the shape of the approximating appendages' tips. It does, however, appear that the shafts are of equal girth. With this in mind I shall assume that the round edges are semi-circles, each rectangle has the same width, and the phallic maxima all touch the curve.
Suppose, then that we were to split a certain area into n little fellas of equal width. The x-coordinates of the left, centre, and right of the rth are as shown in the following diagram.
Using this information, we can investigate the dimensions of this shape in further detail:
The total area of this element is the sum of the rectangle and semi-circle:
This is slightly hamfisted. If we relabel the width as w and simplify we get:
And, should this method work, we claim:
Rather than proving this in general, I'll verify it in a particular case.
We know, from simple Calculus that
If we use the approximation above we get:
In this final line, it's easy to see that as n gets very big, most of the terms get very small. The only exception is the very first term which tends to the required answer.
We can evaluate this expression for various values of n as shown in the following table:
So what can we conclude about numerical integration using this rather subversive method? As you can see from the table, the estimates begin quite small and growth is disappointingly slow. It is, however, inevitable that if what we are plugging in becomes large enough, we will get there in the end.
from matheminutes http://ift.tt/2qyrO3y
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