[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.
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.
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.