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Wikipedia says the trapezium rule is "a technique for approximating the definite integral" (my emphasis).

Isn't the trapezium rule identical with definite integration as the number of strips gets large and the width of each strip gets small?

mjc
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  • Not quite identical, since for a definite (Riemann) integral one uses rectangles as opposed to parallelograms. – G. Chiusole Aug 21 '19 at 10:27
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    @G.Chiusole In the limit they're the same though, no? – It'sNotALie. Aug 21 '19 at 10:30
  • @G. Chiusole Might it then be more accurate to say that "definite integration is a technique for approximating the trapezium rule", since trapezia approximate areas under curves better/more quickly than rectangles? – mjc Aug 21 '19 at 10:31
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    @It'sNotALie. Yes, that is true. A proof is here. – G. Chiusole Aug 21 '19 at 10:35
  • @mjc I wouldn't say so, no. The trapezium rule is something used to quickly approximate the definite integral by hand, whereas the definite integral aims to give the exact value of the integral. In the limit they are the same, yes. I.e. one could just as well define the Riemann integral via trapezoids. See the link in the above comment – G. Chiusole Aug 21 '19 at 10:38
  • If one only does finitely many iterations of refinements i.e. if one only uses finitely many trapeziods/rectangles then, one might say that trapezioids give a better approximation – G. Chiusole Aug 21 '19 at 10:39
  • I think we're on the same page. Thanks for answering. – mjc Aug 21 '19 at 10:42
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    There are some approximation techniques (quadratures) that work great for a low number of points, but become very unstable as the number of points increases. This method is not like that, and so, both Trapezium and rectangle methods can be used to define the Riemann integral as the limit or Riemann sums. I think more complicated methods, like Simpson's can be used for this purpose as well – Yuriy S Aug 21 '19 at 10:56

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Given a function $x\mapsto f(x)\in{\mathbb R}$ $\>(a\leq x\leq b)$ the integral $\int_a^b f(x)\>dx$ is a certain real number. This number has various intuitive descriptions, and is mathematically defined by an involved limiting process. The trapezian rule is a formula for obtaining approximations to this number. It contains a parameter $n\in{\mathbb N}$ or $h>0$ defining the fineness of the intended approximation. If $f$ is integrable over the interval $[a,b]$ then it is usually easy to show that under $\lim_{n\to\infty}$, or $\lim_{h\to0}$, the trapezian sums converge to to the intended integral. But each individual sum is just a finite sum, and is not equal to the integral. The interesting point, however, is to estimate the error when we use a trapezian sum $T_n$ as approximate value for the intended integral.

Christian Blatter
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