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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 $$\int_{a}^{b}y(x)dx:=\lim_{\delta x \to 0} \sum_{x=a}^{b}\frac{y(x)+y(x+1)}{2} \delta x \tag{sum of areas of trapezia}$$?

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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.

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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.

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  • $\begingroup$ 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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