You should already have read or skimmed part 1 and part 2 of this saga. In part 2, we showed the picture of the globe (reproduced here) that purported to show how the aspect ratio of a 10° by 10° box at 60° was 2:1.
Of course, it’s only approximately 2:1, I told the students, because (small pause to get attention) the scale on the globe is changing with every degree. In fact, it’s changing continuously. At this point, the students muttered, “mmm, calculus.” So at least they smelt it coming.
I’d like to be able to say that at this point I turned it over to them: “Indeed! Calculus! We’re trying to find 
But I can’t. Being realistic, it was good that they sensed calculus, but in this unusual context—the sphere and everything—it would have been excruciating. So I scaffolded like my life depended on it. I reproduce the chart we developed last time that shows the dimensions of the 1° boxes.
“Indeed! Calculus! After all, we want
I write the sum on the board, (still ignoring the constant factor, by the way, but that was OK):
“And of course, within each of these 1° sections, it’s changing continuously too. So what should we do?”
Now, almost in unison, “integrate secant,” and a student bravely writes,
Doing the Integral—the way cool way
“And of course, you all know the integral of the secant, right?”
They squirm, but I take them off the hook. “Of course you don’t. Nobody should remember that. But what’s the most practical way to find this integral?”
“A substitution?” they ask. “Do we, like, do it by parts somehow?”
“All good suggestions, but it’s really not obvious what to substitute. And for parts, usually there’s more stuff, right? So you can have a v and a du? What I’m really asking is, what’s a really practical way to find this integral?”
“Look it up?”
“Yeah! How?”
“Google it?”
“That will work, but I want to show you another way.” I pull out my iPhone and hook it up to the projector. “Siri, integrate secant of x.”
“You’re kidding me,” says one kid. Class chatter rises.
“Hmm, let me think…,” says Siri. She shows me an approximation and a link to Wolfram Alpha, which I tap. And this appears:
At this point, they returned to desmos and their data to see if this function actually worked. And it did, to enormous satisfaction throughout the class. That this obscure function, a log of a combination of trig functions, fit these data, was a bit of a miracle, a triumph of actually using calculus.
Actually Finding the Integral
We did not find the integral in class, but in case you care, here is a derivation, which shows just how arcane these can get. This is why you look these up. And why it’s so great that Wolfram Alpha exists, because they give you the answer and the derivation.
It turns out that there is a great substitution! First you multiply top and bottom by 
Now we do a u-substitution, choosing
When you take its derivative, you discover that by some miracle,
Which means that
Which turns out (after even more mind-numbing algebra than it took to find du above) to be equivalent to the horrible formula with the half-angles that Wolfram gave us at first.
Perhaps there is a way to anticipate that that particular substitution would work out so well, but I sure don’t have that kind of insight.
A Best-Case Scenario 
If you have a table of derivatives that includes the derivatives of sec(x) and tan(x), the substitution is not so hard to come up with. d sec(x)/d x = tan(x) sec(x) and d tan(x) = (sec(x))^2 both have a sec term in the derivative (which is what we want to do antiderivative) and adding them gives the tricky d(sec(x) + tan(x))/dx = (sec(x)+tan(x)) sec(x).