Daily Archives: February 25, 2011

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 .

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 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 , , , etc. We looked at where fnInt broke down.

This is what we found out:

The left column is the exponent in . 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 ? And maybe it’ll also work for non-integral values, like ? We’ll see.

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

Today in multivariable calculus, we were talking generally about . Before we embark on evaluating this integral, I wanted kids to guesstimate using their calculators what the value is.

The calculator image showed:

They had a conjecture as to what was going wrong when we expanded the interval… the calculator might be doing a finite number of Riemann Sums, then the width of each rectangle would be large andthe height (especially near the hump near 0) would be small.

Okay I’m describing it terribly… maybe a terrible picture will help.

Good conjecture. Great conjecture, in fact. But I doubted that the TI-83/84 uses Riemann Sums to do fnInt.

It was the end of class, so I sent my kids off with this one charge: investigate how the TI-83/84 calculates integrals, and see if you can’t explain why we’re getting funky answers for a large interval.

I figured I’d pose the question to you, if any of you are calculator saavy…

I wonder if it has to do with the fact that the calculator can only store so many (is it 15?) digits — as part of it?

PS. My very limited research has led me to the fact that the calculator does something called Gauss-Kronrod quadrature, which is a lot of gobbly gook to me right now.