Finding Area Under Curve: Rectangles vs Trapezia

[whatisreality]

Two numerical methods for finding the area under a curve are the trapezium rule, where the area is split into trapezia, and the rectangle rule where you split into rectangles. The rectangle rule has two forms, one where you take the height at the midpoint and one where you take the height of the vertex on the left.

Given the area is split into the same number of smaller shapes, is there ever going to be a case when the rectangles are better than trapezia? I can't think of one! But there must be a case where rectangles are better, or why bother learning the method?

 

[mathman]

Rectangle - height in middle could be better than trapezium.

 

[whatisreality]

Rectangle - height in middle could be better than trapezium.

Even if the height in the middle is better, won't the points on either side negate that?

 

[BvU]

Two numerical methods for finding the area under a curve are the trapezium rule, where the area is split into trapezia, and the rectangle rule where you split into rectangles. The rectangle rule has two forms, one where you take the height at the midpoint and one where you take the height of the vertex on the left.

Given the area is split into the same number of smaller shapes, is there ever going to be a case when the rectangles are better than trapezia? I can't think of one! But there must be a case where rectangles are better, or why bother learning the method?

I don't understand mathman's reply.
To me rectangle is zero'th order and trapezium is first order approximation to the function being integrated. Check the error analysis sections in the links.

I suppose one can concoct a pathological case where (by accident) the simpler method comes out better, but why bother ?

And 'learning' the method is sensible, because it's so evident and uncomplicated.

 

[mathman]

Simple example: \int_0^1 x^2dx = 1/3. One interval. Trapezium est. = 1/2, midpoint est. = 1/4. Midpoint slightly better.

 

[aikismos]

I'm going to go out on a limb and say it's a little more complicated mathematically. I'd wager different types of curves might have an effect on the ranking of which is the most effective method. I was looking at the error analysis of numerical analysis formulas, and it looks like to answer your question (even holding the widths of the divisions the same across methods) requires an examination of the curve as well as the method.

Of course, sometimes less effective is taught because it's simpler, and simpler is easier to learn.

Here seems to be comparison of error terms between rectangular and trapezoidal (See https://en.wikipedia.org/wiki/Newton–Cotes_formulas). Look under the section labeled Open Newton-Cotes Formulae in the last column.