0:00

[MUSIC] We're going to sneak up on a certain fact.

I want to show that pi is less than 22 sevenths.

These are awfully close. Pi is about 3.1416.

And 22 seventh is about 3.1429. Alright?

These are awfully close. And we'll do it with an[UNKNOWN].

I'm just going to make something up, and it'll turn out in retrospect to be

exactly what we need. So the trick's going to be, do the

integral from 0 to 1 of x to the fourth. Times one minus x to the fourth all over

one plus x squared dx. How do I integrate that?

Well, before we actually do the calculation, let's notice something about

the integrand. the denominator here is positive.

And the numerator, well as long as x isn't 0 or 1, the numerator's positive as

well. It's continuous and I'm integrating a

continuous function that's mostly positive except at the end point.

So that means that this integral ends up being positive as well.

We'll need that fact later. The, the fact that the integral's

positive. But let's work now on the integrand.

I'm going to focus first on the numerator.

1 minus x to the fourth. If I just multiply this out I get x to

the fourth minus 4x cubed plus 6x squared minus 4x plus 1.

1:34

And now if I multiply this by x to the fourth, that just adds 4 to each of these

exponents. This will be x to the eighth minus 4 x to

the seventh, plus 6 x to the sixth, minus 4 x to the fifth, plus x to the fourth.

Let's get ready to do some long division. So I'm just going to copy this down.

So, what did I get as the numerator, what was x to the fourth times 1 minus x to

the fourth. Expand it out, was x to the eight minus

4x to the seventh plus 6x to the sixth minus 4x to the fifth.

Plus x to the fourth, and I'm going to take that numerator and I divide it by

one plus x squared, so this we're going to do long division.

I'm dividing this by, I write it as x squared plus one.

Okay, so that's the calculation I want to do.

I want to do this long division problem. Well I gotta put something up here that I

multiply by x squared plus 1 to kill x to the 8th.

And, for that I could use x to the 6th, because x to the 6th times x squared is x

to the eighth. And x to the sixth times 1, which is plus

x to the 6th. And I'm going to subtract that, and what

do I get? Well, the x to the 8th terms cancel, and

I've got a minus 4x to the 7th. 6x to the sixth minus x to the sixth is 5

x to the sixth minus 4 x to the fifth and plus x to the fourth.

Those survive and I'm subtracting anything as soon as copy them down, minus

4 x to the fifth plus x to the fourth. Alright, now you gotta put something up

here that I can multiply by x squared plus 1 to your minus 4 x to the seventh.

but minus 4x to the fifth. because, that times x squared gives me

minus 4x to the seventh and that times 1 gives me ooh, minus 4x to the fifth.

That's Great! So this term dies and this term dies and

let's copy down the terms actually it's 5x to the sixth.

Plus x to the fourth. Now what do I put up here to multiply by

this to kill this? plus five x to the fourth will do it,

because that times x squared gives me five x to the sixth, and five x to the

fourth times one gives me plus 5x to the fourth.

And remember, I'm subtracting this. So that kills this first term.

But here I've got what minus 4x to the fourth.

I'm going to write that over here. So, minus 4x to the fourth is all that's

left. I've gotta put something up here to

multiply by this to kill this. How about minus 4x squared, because that

times this gives me minus 4x to the fourth.

And then minus 4x squared. I'm subtracting this so this becomes just

4x squared. I've got to put something here to

multiply by this to kill this. Well plus 4 will do it.

Plus 4 times that gives me 4x squared plus 4.

And when I subtract this I get minus four left over and that's my remainder.

Now we can rewrite the integral. So in light of the expansion of the

numerator and the long division after I divided by 1 plus x squared.

This is the integral from 0 to 1. Of, well, what did I get when I divided?

Was x to the sixth minus four x to the fifth plus five x to the fourth minus

four x squared. Plus four and then I had a remainder so

minus four over one plus x squared. This is an integral I can do.

Okay this is x to the seventh over seven. That is the anti-derivative x to the

sixth. the anti-derivative here is four x to the

sixth over six. Plus an antiderivative here.

It's just x to the fifth, antiderivative here is minus 4 x to the third over 3.

Antiderivative of 4 is just plus 4x, and an antiderivative of 4 over 1 plus x

squared, well the antiderivative of 1 over 1 plus x squared's arctan.

So this is minus 4, orcten x and I'm integrating from 0 to 1, so I'm just

going to evaluate this from 0 to 1. Now, what do I get when I evaluate this?

When I plug in 0, I get 0, 0, 0, 0, 0. And arctan of 0 is 0.

So, I end up just subtracting 0. So I only have to just evaluate this at 1

to figure out what this integral is equal to.

So when I plug in 1, I get 1 seventh. Minus 4 sixths plus one minus 4 thirds

plus 4 minus 4 times, and what's arctan of 1, well arctan of is pi over 4, let's

simplify a bit. So, I've got 1 seventh, and I'll put all

these with a common denominator of three. So, minus 2 thirds plus 3 thirds minus 4

thirds plus 12 thirds, and then minus 4 times pi over 4 is just minus pi.

6:47

Well now what do I have? let's see here.

So I've got minus 2 minus 4 that's minus 6 thirds, 3 plus 12, that's 15 thirds

minus 6 thirds. That gives me 9 thirds leftover, and then

also minus pi. So I've got a seventh.

Plus, well 9 thirds is just 3 minus pi. Well 1 seventh plus 3 that's 22 sevenths

minus pi. Here's the significance of this, remember

back to the beginning. Well way back at the beginning I pointed

out that this integral is positive. So what have we shown?

So we've shown that this, the value of the integral is positive.

In other words, that 22 sevenths is bigger than pi.

So this is really a fun example. We've got all these pieces coming

together. We've got the derivative of arctan, long

division of polynomials, the fundamental theorem of calculus.

All these forces combining to convince us that 22 sevenths is bigger than pi.

So if you want an entertaining challenge, here is an even more complicated integral

to evaluate. The interval x goes from 0 to 1 of x to

the eighth of 1 minus x to the eighth. Times 25 plus 816 times x squared and all

of that over 3164 times one plus x squared.

And the answer that you get here is pretty entertaining.