Need Help With a Funky Integral Problem

Factual Questions
1

I am a high school calculus teacher. I’m working through some review problems that I’m about to give my students and I’m stumped by one of them.
If F’(x)=sqrt(1+x^3) and F(1) = 5, then F(3) = ?

The answer is 11.230 but I can’t figure out how to get there.

My initial thought was to find the integral of F’(x) to get F(x), use F(1) = 5 to solve for C, then evaluate the new function for F(3). But this technique falls apart since you can’t integrate sqrt(1+x^3) directly without x^2 on the outside.
Any ideas?

2

Typo.

Either that or maybe one is supposed to use direction fields to estimate the answer if that was a topic you covered.

3

Yeah, my gut reaction is that that’s going to work out to be some hypergeometric function. (fires up Mathematica) Ah, an elliptic integral with complex arguments. Even better.

The fact that the quoted answer is 11.230 rather than some exact result (in terms of π, e, sines, cosines, square roots, rational fractions, etc.) makes me think that some kind of numerical technique is desired and/or required.

4

I’m thinking typo as well, Lance. I only have the answer and not the solution key to check on that though. We have gone through slope fields but the problem came from a chapter that is long before that topic is even introduced.

How would one go about a numerical technique for solving this? The students know all about Riemann Sums and such for evaluating definite integrals but this one doesn’t have bounds - unless they need to be pulled somehow from the initial conditions.

5

F(3) - F(1) = the integral of F’(x) taken from 1 to three (by FTC).

We can solve this for F(3) and use the given F(1) and an approximation of the integral to approximate F(3).

This does in fact yield F(3) = 11.29959.

6

Agreed. This is clearly supposed to be an approximation problem, as those are the only problems in which answers would be quoted in decimals. The fact that it’s given to so many significant figures indicates that it’s likely supposed to be done by calculator. The structure of the problem forces students to not just type it into the calculator and report the answer; they have to think about what information they need out of the calculator and how to use that information to answer the question.

Do they still allow the same graphing calculators on AP exams that they did when I took them 10 years ago? I have to imagine that there’s handhelds that are significantly more powerful now.

7

I just realized I typoed the answer in my previous post. I actually calculated:

F(3) = 11.229959… which does round to the desired answer.

8

Thank you so much!! This makes perfect sense. It’s twisty and wonderful!

I just knew that someone on here would be able to figure this out. :slight_smile: