fnInt reprise

My last post was about how the TI-83/84 calculates integrals (how fnInt works), and how it messes up for when you have large intervals.

I just came from my Multivariable Calculus class, where each student had done some thinking about it. One investigated the Gauss-Kronrod quadrature. A couple others played around with fnInt and came up with some bounds for when fnInt was good and when fnInt was bad for our function f(x)=e^{-x^2}.

What we did today was to start investigating fnInt in a different way. (Yeah, my goal was to start triple integrals today… but this was way more exciting in the moment…)

We looked at \int_1^{\infty} \frac{1}{x^2}dx and used fnInt to calculate it.

It turns out that fnInt goes crazy and fails to be a good estimator at a particular large interval.

So we continued looking at \frac{1}{x^3}\frac{1}{x^4}\frac{1}{x^5}, etc. We looked at where fnInt broke down.

This is what we found out:

The left column is the exponent in \frac{1}{x^n}. The right column is the last integer you can integrate (using fnInt) to so that doesn’t give a terrible estimation of the area. (Recall we’re integrating from 1 onwards, not from 0.)

My kids are going to go home and see what they can make of this data. We hope we can use it to come up with a prediction for where fnInt will go awry for estimating the area for something like \frac{1}{x^{43}}? And maybe it’ll also work for non-integral values, like \frac{1}{x^{3.23}}? We’ll see.

…Hopefully we’ll start on triple integrals soon, though…


  1. See my comment on tolerance in your previous post. Increasing the tolerance for each numerical integration should increase the maximum domain that still yields a reasonable result. For x^(-21), fnInt worked up to [1,530], yielding an answer of 1/2 before the calculator decided it wasn’t worth its time for 531 and called it 9.86E-11.

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