# 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…