I've been familiar with integration, its motivation and its methods for a while know, but this question has only just entered my mind:

When we integrate to find the area under a curve, why do we split up the curve into infinitesimally-thin rectangles rather than trapezia?

Surely, trapezia are more accurate to work with.

My motivation for this question is that a popular method for numerical integration, the trapezium rule, as its name suggests, splits the curve into arbitrarily-thin trapezia as opposed to rectangles.

So why is the integral defined as $$\int_{a}^{b}y(x)dx:=\lim_{\delta x \to 0} \sum_{x=a}^{b}y(x)\delta x \tag{sum of areas of rectangles}$$ rather than what appears to be the more-accurate $$\int_{a}^{b}y(x)dx:=\lim_{\delta x \to 0} \sum_{x=a}^{b}\frac{y(x)+y(x+1)}{2} \delta x \tag{sum of areas of trapezia}$$?


If we are using "infinitely thin" shapes, rectangles and trapezia are equally accurate. For numerical integration, however, you will be using shapes of a definite width, and in this case, generally speaking, trapezia will give a more accurate answer. For some idea of what determines the accuracy of the trapezoidal approximation, you could start here.


Both limits are the same for integrable functions, so it really makes no difference which one you take...

Also, as far as I know, the integral is defined as the limit of Riemann sums for which you can take any division of $[a,b]$ into as many subintervals as you want, so long as their longest one limits to $0$, and then take any set of values, one from each interval. So your definition is lacking.

  • $\begingroup$ So is it that we only use the first definition (as opposed to the second) because the first is simpler? $\endgroup$ – beep-boop Jul 14 '14 at 13:24
  • $\begingroup$ @alexqwx As I wrote in my edit, the "real" definition is more complicated than what you wrote. $\endgroup$ – 5xum Jul 14 '14 at 13:26
  • $\begingroup$ See the wikipedia article for the proper definition: en.wikipedia.org/wiki/Riemann_integral#Riemann_integral $\endgroup$ – 5xum Jul 14 '14 at 13:27
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    $\begingroup$ @alexqwx Both definitions converge to the same value. The only difference really is the rate of convergence, which makes the trapezoidal rule more efficient for numerical algorithms. $\endgroup$ – David H Jul 14 '14 at 13:27

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