Rectangle Method for approximating an integral has been rediscovered

[Dembadon]

"Rectangle Method" for approximating an integral has been rediscovered!

http://care.diabetesjournals.org/content/17/2/152.abstract

A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves
Mary M Tai, MS, EDD

+ Author Affiliations

Obesity Research Center, St. Luke's-Roosevek Hospital Center New York Department of Nutrition, New York University New York, New York
Address correspondence and reprint requests to Mary M. Tai, MS, EdD, Department of Nutrition, New York University, Education Building #1077, 35 West 4th Street, New York, NY 10012.

Abstract

OBJECTIVE To develop a mathematical model for the determination of total areas under curves from various metabolic studies.

RESEARCH DESIGN AND METHODS In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method Gess than ±0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin.

RESULTS Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies.

CONCLUSIONS The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.

Received February 18, 1993.
Accepted September 23, 1993.
Copyright © 1994 by the American Diabetes Association

Is this some kind of joke? Has anyone else seen this article before?

This is what I felt like after reading it:

 

[Dickfore]

She called it after herself. That's modest enough.

 

[Dickfore]

I found this:

http://care.diabetesjournals.org/content/17/10/1225.2.full.pdf (it's a PDF)

EDIT:
It's pretty popular on the Internet:
Reinventing the wheel

 

[Curious3141]

OK, I'm going to publish another paper employing Simpson's Rule (but not call it that), prove it's better or faster-converging than her approximation, and name it after myself.

Seriously, this is embarrassing. Well, she and the referees of the journal should be embarrassed.

 

[Jimmy Snyder]

This reminds me of a story I once heard of an article on economics that was published in some journal. A mathematician wrote into point out a simple typo in one of the equations. I don't remember the name of the mathematician, but say it was Smith. The corrected equation then came to be known as Smith's law.

 

[Drakkith]

I'm not sure I get it. Doesn't calculus do this already? (My knowledge of calculus comes from reading part of a Calculus for Dummies book)

 

[AlephZero]

Why should it be surprising that nutritionists don't know calculus? Hey, you can even study physics without calculus these days, except it's more politically correct to call it "algebra-based physics" not "calculus-free physics".

At least what she did was right. I've seen much worse nonsense perpetrated by people who shouldn't have been allowed anywhere near inventing numerical methods.

 

[Astronuc]

Hmmm - I remember using the technique in 1974-1975 - during a calculus class in high school. We also studied the trapezoidal and Simpson's rules - and the Riemann sum. Later at university (70s and 80s), we delved more into the theory behind quadrature as part of a course on numerical methods which included various quadrature, or numerical integration, techniques.

 

[BobG]

I would have been more impressed if she had reinvented this method:

http://www.ehow.com/how_6209482_calculate-acreage-map.html

She could have determined the area of the entire graph and weighed it.

Then she could have cut out the area under the curve and weighed that.

Simply brilliant!

 

[Dickfore]

I would have been more impressed if she had reinvented this method:

http://www.ehow.com/how_6209482_calculate-acreage-map.html

She could have determined the area of the entire graph and weighed it.

Then she could have cut out the area under the curve and weighed that.

Simply brilliant!

Question: How does one plot the curve exactly according to its functional form?

 

[BobG]

Question: How does one plot the curve exactly according to its functional form?

So that a computer can search through its database to find similar shapes?

That's a pretty tough task (and one I've never done myself, even though I've used the results of that type of analysis). http://vis.uky.edu/~cheung/courses/ee639_fall04/readings/shapeReview.pdf

It's a lot tougher than just comparing the areas unders the curves.

But analyzing the shape itself can be pretty important for a lot of tasks, such as determining orbital perturbations on satellites due to oblateness of the Earth, "bulges" around the equator, etc. with Legendre polynomials being most common method for geodysey (Earth geoid)

 

[Reptillian]

Somebody needs to teach this woman about Runge-Kutta.

 

[256bits]

Is there a limit to the maximum number of times New York can be repeated in the address?

 

[Q_Goest]

It's worse than you think! (Read the paper)

Apparently, Mary Tai doesn't know how to calculate the area of a trapezoid (and thus employ the trapezoidal rule). Instead, Mary calculates the area of a rectangle (Simpson's rule) and then modifies it by ADDING the area of the remaining triangle.

So yea, this is an original work. I can't remember calculus courses teaching this particular method!

 

[Dickfore]

Somebody needs to teach this woman about Runge-Kutta.

Somebody needs to teach you about quadrature.

 

[Dickfore]

So that a computer can search through its database to find similar shapes?

That's a pretty tough task (and one I've never done myself, even though I've used the results of that type of analysis). http://vis.uky.edu/~cheung/courses/ee639_fall04/readings/shapeReview.pdf

It's a lot tougher than just comparing the areas unders the curves.

But analyzing the shape itself can be pretty important for a lot of tasks, such as determining orbital perturbations on satellites due to oblateness of the Earth, "bulges" around the equator, etc. with Legendre polynomials being most common method for geodysey (Earth geoid)

So, in conclusion, your method fails.

 

[AlephZero]

Somebody needs to teach this woman about Runge-Kutta.

Somebody needs to teach you about quadrature.

Solving a differential equation (with automatic error estimation and step size control) can be an excellent way to do numerical quadrature.

 

[someGorilla]

I guess that paper has been referenced already a lot of times. Nice way to increase your number of citations! While giving people a good laugh, what more do you want?
And who knows, she might patent the method. It's not new? Of course it's new, this is specifically for metabolic curves. There's a lot of patents like that, like the sled for carcasses.