The trapezoidal rule for numerical integration is based on the idea that when we partition our larger interval into subintervals, we can approximate the area over each subinterval by calculating the area of the trapezoid formed by connecting the value of the function at the left and right endpoints of the subinterval with a straight line.(adsbygoogle = window.adsbygoogle || []).push({});

In all of the calculus textbooks I was looking through, this is only illustrated using a strictly positive function. When you start looking at a function that is positive and negative, there is no guarantee that over a given subinterval you even get a trapezoid (see the attached image). In that image there are 5 subintervals, and the middle one certainly doesn't yield a trapezoid. If we calculate the actual area over that subinterval it is going to be two triangles, and yield something quite different from what the area of a trapezoid would have been.

I don't know a whole lot about numerical analysis, but I'm guessing this would be one additional source of error in this method? Although you are calculating the area for a trapezoid over each subinterval, in some cases the actual (approximated) area you're dealing with isn't even a trapezoid. I don't know if there's much more to say about this other than that I found it very curious, but I would love to hear the perspective of someone with a stronger numerical analysis background on this issue.

Regards,

-Xaenn

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# Trapezoidal Rule for numerical integration

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