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PROFESSOR: Today I want to get
started by correcting a mistake

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that I made last time.
 

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And this was mistaken
terminology.

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I said that what we were
computing, when we computed

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in this candy bar example
was energy and not heat.

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But it's both.
 

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They're the same thing.
 

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And in fact, energy, heat
and work are all the

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same thing in physics.
 

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I was foolishly considering the
much more, what am I trying to

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say, the intuitive feeling of
heat is just being the

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same as temperature.
 

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But in physics, usually heat is
measured in calories and energy

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can be in lots of things.
 

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Maybe kilowatt-hours or ergs,
these are various of the units.

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And work would be in
things like foot-pounds.

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That is, lifting some
weight some distance.

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And the amount of force
you have to apply.

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And these all have
conversions between them.

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They're all the same quantity,
in different units.

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OK, so these are
the same quantity.

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Different units.
 

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So that's about as much physics
as we'll do for today.

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And sorry about that.
 

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Now, the example that I was
starting to discuss last time

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and then I'm going to carry
out today was this

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dartboard example.
 

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We have a dartboard, which
is some kind of target.

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And we have a person, your
little brother, who's

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standing over there.
 

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And somebody is throwing darts.
 

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And the question is, how
likely is he to be hit.

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So I want to describe to you
how we're going to make this

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problem into a math problem.
 

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Yep.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: What topic is
this that we're going over.

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We're going over an example.
 

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Which is a dartboard example.
 

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And it has to do
with probability.

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OK.
 

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So what is the probability
that this guy, your little

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brother, gets hit by a dart.
 

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Now, we have to put some
assumptions into this

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problem in order to
make it a math problem.

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And I'm really going to
try to make them pretty

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realistic assumptions.
 

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So the first assumption is
that the number of hits is

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proportional to ce ^ - r ^2.
 

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So that's actually a kind
of a normal distribution.

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That's the bell curve.
 

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But as a function
of the radius.

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So this is the assumption
that I'm going to

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make in this problem.
 

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And a problem in probability
is a problem of the ratio

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of the part to the whole.
 

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So the part is where this
little guy is standing.

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And the whole is all the
possible places where

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the dart might hit.
 

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Which is maybe the whole
blackboard or extending beyond,

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depending on how good an aim
you think that this

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older child has.
 

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So, if you like, the part
is, we'll start simply.

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I mean, this doesn't sweep
all the way around.

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But we're going to talk
about some section.

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Like this.
 

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Where this is some radius r1,
and this other radius is

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some longer radius, r2.
 

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And the part that we'll
first keep track of is

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everything around there.
 

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That's not very well-centered,
but it's supposed to be

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two concentric circles.
 

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Maybe I should fix a bit
so that it's a little

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bit easier to read here.
 

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So here we go.
 

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So here I have radius r1,
and here I have radius r2.

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And then the region in between
is what we're going to

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try to keep track of.
 

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So I claim that we'll be able
to get, so this is what I'm

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calling a part, to
start out with.

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And then we'll take a section
of it to get the place where

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the person is standing.
 

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Now, I want to take a
side view of e ^ - r ^2.

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The function that
we're talking about.

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Again, that's the bell curve.
 

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And it sort of looks like this.
 

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This is the top value,
this is now the r axis.

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And this is up, or at least, so
you should think of this in

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terms of the fact that the
horizontal here is the

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plane of the dartboard.
 

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And the vertical is measuring
how likely it is that there

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will be darts piling up here.
 

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So if they were balls tumbling
down or something else

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falling on, many of them
would pile up here.

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Many fewer of them would be
piling up farther away.

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And the chunk that we're
keeping track of is the

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chunk between r1 and r2.
 

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That's the corresponding
region here.

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And in order to calculate this
part, we have to calculate

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this volume of revolution.
 

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Sweeping around.
 

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Because really, in
disguise, this is a ring.

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This is a side view,
it's really a ring.

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Because we're rotating it
around this axis here.

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So we're trying to figure
out what the total

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volume of that ring is.
 

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And that's going to be our
weighting, our likelihood for

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whether the hits are occurring
in this section or in this

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ring versus the rest.
 

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To set this up, I remind
you we're going to use

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the method of shells.
 

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That's really the only one
that's going to work here.

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And we want to integrate
between r1 and r2.

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And the range here is that
because this is, if you like,

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a solid of revolutions.
 

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So the variable r is the same
as what we used to call x.

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And it's ranging between
r1 and r2, and then we're

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sweeping it around.
 

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And the circumference of
a little piece, so at a

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fixed distance r, here,
the circumference is

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going to be 2 pi r.
 

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So 2 pi r is the circumference.
 

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And then the height is
the height of that

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green stem there.
 

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That's e ^ - r ^2.
 

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And then we're multiplying by
the thickness, which is dr. So

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the thickness of the green is
dr. The height of this little

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green guy is e ^ - r^2, and the
circumference is e pi times the

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radius, when we sweep
the circle around.

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Question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: So the question
is, why is we're interested

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in not this pink area.
 

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And the reason is an
interpretation of what

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I meant by this.
 

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What I meant by this is that if
you wanted to add up what the

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likelihood is that this thing
will be here versus here, I

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want it to be, really, the
proportions are the number

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of hits times the, if you
like, I wanted it d area.

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That's really what
I meant here.

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The number of hits
in a little chunk.

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So little, maybe I'll
call it delta a.

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A little chuck.
 

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Is proportional to the
chunk times that.

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So there's already
an area factor.

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And there's a height.
 

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So there are a total of
three dimensions involved.

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There's the area and
then this height.

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So it's a matter of the, what
I was given, what I intended

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to say the problem is.
 

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Yeah, another question.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Yeah, yeah.
 

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Exactly.
 

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The height is c times
that, whatever this c is.

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In fact, we don't know what the
c is, but because we're going

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to have a part and a whole,
we'll divide, the c

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will always cancel.
 

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So I'm throwing the c out.
 

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I don't know what it is, and
in the end it won't matter.

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That's a very
important question.

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Yes.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Say it again?
 

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STUDENT: PROFESSOR:
 

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PROFESSOR: So what I
mean is the number of

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hits in some chunk.
 

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That is, suppose you imagine,
the question is, what does

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this left-hand side mean.
 

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That right?
 

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Is that the question
that's you're asking?

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When I try to understand
what the distribution of

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dartboard hits is, I should
imagine my dartboard.

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And very often there'll
be a whole bunch of

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holes in some places.
 

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And fewer holes else.
 

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I'm trying to figure out
what the whole distribution

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of those marks is.
 

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And so some places will
have more hits on them

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and some places will
have fewer hits on them.

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And so what I want to
measure is, on average,

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the number of hits.
 

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So this would really be
some, this constant of

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proportionality is ambiguous
because it depends on

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how many times you try.
 

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If you throw a thousand
times, it'll be much

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more densely packed.
 

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And if you have only a hundred
times it'll be fewer.

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So that's where this
constant comes in.

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But given that you have a
certain number of times that

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you tried, say, a thousand
times, there will be a whole

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bunch more piled in the middle.
 

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And fewer and fewer as
you get farther away.

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00:12:12 --> 00:12:15
Assuming that the person's
aim is reasonable.

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So that's what we're saying.
 

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So we're thinking in terms
of the person's always

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aiming for the center.
 

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So it's most likely that the
person will hit the center.

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But on the other hand, it's
a fallible person, so

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the person may miss.
 

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And so it's less and less
likely as you go farther out.

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And we're just counting how
many times this gets hit, how

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many time this and so on.
 

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In proportion to the area.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Yeah. r1
and r2 are arbitrary.

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00:12:42 --> 00:12:44
We're going to make this
calculation in general.

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00:12:44 --> 00:12:46
We're going to calculate what
the likelihood is that we

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hit any possible band.
 

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And I want to leave those
as just letters for now.

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00:12:53 --> 00:12:54
The r1 and the r2.
 

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00:12:54 --> 00:12:57
Because I want to be able
to try various different

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00:12:57 --> 00:12:57
possibilities.
 

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00:12:57 --> 00:13:03
STUDENT: [INAUDIBLE]
 

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00:13:03 --> 00:13:06
PROFESSOR: Say it again., why
do we have to take volume.

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00:13:06 --> 00:13:08
So this is what we were
addressing before.

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00:13:08 --> 00:13:13
It's a volume because it's
number of hit per unit area.

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So there's a height, that
is, number of hits.

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00:13:17 --> 00:13:20
And then there's an area and
the product of those is, so

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this is, if you like, a
histogram of the

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00:13:23 --> 00:13:25
number of hits.
 

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00:13:25 --> 00:13:26
But this should be
measured per area.

231
00:13:26 --> 00:13:29
Not per length of r.
 

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00:13:29 --> 00:13:32
Because on the real diagram,
it's going all the way around.

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00:13:32 --> 00:13:36
There's a lot more area to
this red band than just

234
00:13:36 --> 00:13:41
the distance implies.
 

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00:13:41 --> 00:13:42
OK.
 

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00:13:42 --> 00:13:48
So, having discussed the setup,
this is a pretty standard setup

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00:13:48 --> 00:13:49
-- oh, one more question.
 

238
00:13:49 --> 00:13:49
Yes.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Yeah.
 

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00:14:14 --> 00:14:17
So the question is, why
is this a realistic.

242
00:14:17 --> 00:14:21
Why is this choice of function
here e to the minus r squared

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00:14:21 --> 00:14:25
a realistic choice of
function for the darts.

244
00:14:25 --> 00:14:30
So I can answer this
with an analogy.

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00:14:30 --> 00:14:34
When people were asking
themselves where the V-2

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00:14:34 --> 00:14:39
rockets from Germany hit
London, they used this model.

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It turned out to be the one
which was the most accurate.

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So that gives you an idea
that this is actually real.

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The question, this makes it
look like people are masters.

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That is, that they'll all
hit in the center more

251
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often than elsewhere.
 

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But that's actually
somewhat deceptive.

253
00:15:03 --> 00:15:08
There's a difference between
the mode, that is, the most

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00:15:08 --> 00:15:13
likely spot, and what
happens on average.

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00:15:13 --> 00:15:17
So in other words, the single
most likely spot is the center.

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00:15:17 --> 00:15:20
But there's rather little area
in here, and in fact the

257
00:15:20 --> 00:15:24
likelihood of hitting that
is some little tiny bit.

258
00:15:24 --> 00:15:25
In here.
 

259
00:15:25 --> 00:15:29
In fact, you're much more
likely to be out here.

260
00:15:29 --> 00:15:32
So if you take the total of the
volume, you'll see that much of

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00:15:32 --> 00:15:34
the volume is contributed
from out here.

262
00:15:34 --> 00:15:37
And in fact, the person hits
rather rarely near the center.

263
00:15:37 --> 00:15:42
So this is not a
ridiculous thing to do.

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00:15:42 --> 00:15:45
If you think of it in terms of
somebody's aiming at the center

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00:15:45 --> 00:15:47
but there's some random thing
which is throwing the person

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00:15:47 --> 00:15:51
off, then there is likely to be
to left or to the right, or

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they might even get lucky and
all those errors cancel

268
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themselves and they happen
to hit pretty much

269
00:15:56 --> 00:15:59
near the center.
 

270
00:15:59 --> 00:16:00
Yeah, another question.
 

271
00:16:00 --> 00:16:05
STUDENT: [INAUDIBLE]
 

272
00:16:05 --> 00:16:06
PROFESSOR: How does the little
brother come into play?

273
00:16:06 --> 00:16:09
The little brother is going to
come into play as follows.

274
00:16:09 --> 00:16:11
I'll tell you in advance.
 

275
00:16:11 --> 00:16:15
So the thing is, the little
brother was not so stupid as to

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00:16:15 --> 00:16:19
stand in front of the target.
 

277
00:16:19 --> 00:16:22
I know.
 

278
00:16:22 --> 00:16:26
He stood about twice the
radius of the target away.

279
00:16:26 --> 00:16:29
And so, we're going to
approximate the location

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00:16:29 --> 00:16:32
by some sector here.
 

281
00:16:32 --> 00:16:35
Which is just going to be some
chunk of one of these things.

282
00:16:35 --> 00:16:38
We'll just break
off a piece of it.

283
00:16:38 --> 00:16:39
And that's how we're
going to capture.

284
00:16:39 --> 00:16:42
So the point is, the
target is here.

285
00:16:42 --> 00:16:46
But there is the possibility
that the seven-year-old who's

286
00:16:46 --> 00:16:48
throwing the darts actually
missed the target.

287
00:16:48 --> 00:16:50
That actually happens a lot.
 

288
00:16:50 --> 00:16:54
So, does that answer
your question?

289
00:16:54 --> 00:16:57
Alright. are we
ready now to to do?

290
00:16:57 --> 00:16:57
One more question.
 

291
00:16:57 --> 00:17:05
STUDENT: [INAUDIBLE]
 

292
00:17:05 --> 00:17:08
PROFESSOR: I'm giving
you this property here.

293
00:17:08 --> 00:17:11
I'm telling you that this is
what's called a mathematical

294
00:17:11 --> 00:17:13
model, when you give somebody
something like this.

295
00:17:13 --> 00:17:16
In fact, that requires
further justification.

296
00:17:16 --> 00:17:19
It's an interesting issue.
 

297
00:17:19 --> 00:17:19
Yeah.
 

298
00:17:19 --> 00:17:26
STUDENT: [INAUDIBLE]
 

299
00:17:26 --> 00:17:27
PROFESSOR: I'm giving
it to you for now.

300
00:17:27 --> 00:17:30
And it's something which
really has to be justified.

301
00:17:30 --> 00:17:32
In certain circumstances
it is justified.

302
00:17:32 --> 00:17:37
But, OK.
 

303
00:17:37 --> 00:17:43
So anyway, here's our part.
 

304
00:17:43 --> 00:17:48
This is going to be our chunk,
for now, that we're going to

305
00:17:48 --> 00:17:50
estimate the the importance.
 

306
00:17:50 --> 00:17:56
The relative importance of.
 

307
00:17:56 --> 00:18:00
And now, this is something
whose antiderivative we can

308
00:18:00 --> 00:18:03
just do by substitution
or by guessing.

309
00:18:03 --> 00:18:07
It's just - pi e ^ - r^2.
 

310
00:18:07 --> 00:18:11
If you differentiate that, you
get a - 2r, which cancels

311
00:18:11 --> 00:18:12
the minus sign here.
 

312
00:18:12 --> 00:18:15
So you get 2pi re ^ - r ^2.
 

313
00:18:15 --> 00:18:18
So that's the antiderivative.
 

314
00:18:18 --> 00:18:24
And we're evaluating
it at r1 and r2.

315
00:18:24 --> 00:18:28
So with the minus sign that's
going to get reversed.

316
00:18:28 --> 00:18:34
The answer is going to be pi (
e ^ - r1^2, that's the bottom

317
00:18:34 --> 00:18:39
one. - e ^ - r2 ^2),
that's the top.

318
00:18:39 --> 00:18:45
So this is what our
part gives us.

319
00:18:45 --> 00:18:48
And, more technically, if you
wanted to multiply through by

320
00:18:48 --> 00:18:50
c, it would be c times this.
 

321
00:18:50 --> 00:18:55
I'll say that in just a second.
 

322
00:18:55 --> 00:18:58
OK, now I want to work.
 

323
00:18:58 --> 00:19:04
So, if you like, the part is
equal to maybe even I'll call

324
00:19:04 --> 00:19:12
it c pi (e^ -r1 ^2
- e ^ - r2 ^2).

325
00:19:12 --> 00:19:14
That's what it really is
if I put in this factor.

326
00:19:14 --> 00:19:17
So now there's no prejudice as
to how many attempts we make.

327
00:19:17 --> 00:19:20
Whether it was a thousand
attempts or a million

328
00:19:20 --> 00:19:22
attempts at the target.
 

329
00:19:22 --> 00:19:26
Now, the most important second
feature here of these kinds of

330
00:19:26 --> 00:19:29
modeling problems is, there is
always some kind

331
00:19:29 --> 00:19:30
of idealization.
 

332
00:19:30 --> 00:19:33
And the next thing that I want
to discuss with you is the

333
00:19:33 --> 00:19:36
interpretation of the whole.
 

334
00:19:36 --> 00:19:41
That is, what's the family
of all possibilities.

335
00:19:41 --> 00:19:47
And in this case, what I'm
going to claim is that the

336
00:19:47 --> 00:19:52
reasonable way to think of the
whole is it's that r can range

337
00:19:52 --> 00:19:55
all the way from 0 to infinity.
 

338
00:19:55 --> 00:19:57
Now, you may not like this.
 

339
00:19:57 --> 00:20:01
But these are maybe my first
and third favorite number, my

340
00:20:01 --> 00:20:03
second favorite number being 1.
 

341
00:20:03 --> 00:20:07
So infinity is a really
useful concept.

342
00:20:07 --> 00:20:12
Of course, it's nonsense in
the context of the darts.

343
00:20:12 --> 00:20:16
Because if you think of the
basement wall where the kid

344
00:20:16 --> 00:20:20
might miss a target, he'd
probably hit the wall.

345
00:20:20 --> 00:20:22
He's probably not going to hit
one of the walls to the right,

346
00:20:22 --> 00:20:25
and anyway he's certainly not
going to hit over there.

347
00:20:25 --> 00:20:29
So there's something artificial
about sending the possibility

348
00:20:29 --> 00:20:32
of hits all the way
out to infinity.

349
00:20:32 --> 00:20:37
On the other hand, the shape of
this curve is such that the

350
00:20:37 --> 00:20:39
real tail ends here, because
of this exponential

351
00:20:39 --> 00:20:40
decrease, are tiny.
 

352
00:20:40 --> 00:20:42
And that's negligible.
 

353
00:20:42 --> 00:20:46
And the point is that actually
the value, if you go all the

354
00:20:46 --> 00:20:50
way out to infinity, is the
easiest value to calculate.

355
00:20:50 --> 00:20:52
So by doing this, I'm
idealizing the problem but I'm

356
00:20:52 --> 00:20:55
actually making the numbers
come out much more cleanly.

357
00:20:55 --> 00:20:58
And this is just always
done in mathematics.

358
00:20:58 --> 00:21:00
That's what we did when we
went from differences to

359
00:21:00 --> 00:21:02
differentials, to
differentiation and

360
00:21:02 --> 00:21:04
infinitesimals.
 

361
00:21:04 --> 00:21:06
So we like that.
 

362
00:21:06 --> 00:21:08
Because it makes things
easier, not because it

363
00:21:08 --> 00:21:10
makes things harder.
 

364
00:21:10 --> 00:21:12
So we're just going to
pretend the whole is

365
00:21:12 --> 00:21:13
from 0 to infinity.
 

366
00:21:13 --> 00:21:15
And now let's just
see what it is.

367
00:21:15 --> 00:21:21
It's c pi times the starting
place is e^ - 0 ^2.

368
00:21:21 --> 00:21:24
That's r1, right, this is the
role that r1 plays is this,

369
00:21:24 --> 00:21:26
and the r2 is this value.
 

370
00:21:26 --> 00:21:30
Minus e ^ - infinity ^2.
 

371
00:21:30 --> 00:21:33
Which is that negligibly
small number, 0.

372
00:21:33 --> 00:21:36
So this is just c pi.
 

373
00:21:36 --> 00:21:39
Because this number is 1,
and this other number is 0.

374
00:21:39 --> 00:21:48
This is just (1 - 0), in
the parentheses there.

375
00:21:48 --> 00:21:52
And now I can tell you from
these two numbers what

376
00:21:52 --> 00:21:54
the probability is.
 

377
00:21:54 --> 00:21:59
The probability that we landed
on the target in a radius

378
00:21:59 --> 00:22:05
between r1 and r2, so that's
this annulus here, is the ratio

379
00:22:05 --> 00:22:12
of the part to the whole.
 

380
00:22:12 --> 00:22:16
Which in this case just
cancels the c pi.

381
00:22:16 --> 00:22:21
So it's (e ^ - r1
^2 - e ^ - r2^2).

382
00:22:21 --> 00:22:25
There's the formula
for the probability.

383
00:22:25 --> 00:22:28
So the c canceled and
the pi canceled.

384
00:22:28 --> 00:22:32
It's all gone.
 

385
00:22:32 --> 00:22:34
Now, again, let me just
emphasize the way this

386
00:22:34 --> 00:22:35
formula is supposed to work.
 

387
00:22:35 --> 00:22:43
The total probability of every
possibility here is supposed

388
00:22:43 --> 00:22:45
to be set up to be 1.
 

389
00:22:45 --> 00:22:48
This is some fraction of 1.
 

390
00:22:48 --> 00:22:51
If you like, it's a percent.
 

391
00:22:51 --> 00:22:51
Yes.
 

392
00:22:51 --> 00:22:57
STUDENT: [INAUDIBLE]
 

393
00:22:57 --> 00:22:59
PROFESSOR: This one is giving,
the question is, doesn't this

394
00:22:59 --> 00:23:00
just give the probability
of the ring?

395
00:23:00 --> 00:23:04
This gives you the probability
of the ring, but this is

396
00:23:04 --> 00:23:05
a very, very wide ring.
 

397
00:23:05 --> 00:23:08
This is a ring starting with
0, nothing, on the inside.

398
00:23:08 --> 00:23:09
And then going all the way out.
 

399
00:23:09 --> 00:23:11
So that's everything.
 

400
00:23:11 --> 00:23:13
So this corresponds
to everything.

401
00:23:13 --> 00:23:22
This corresponds to a ring.
 

402
00:23:22 --> 00:23:24
So now, let's see.
 

403
00:23:24 --> 00:23:26
Where do I want
to go from here.

404
00:23:26 --> 00:23:29
So in order to make progress
here, I still have to give you

405
00:23:29 --> 00:23:31
one more piece of information.
 

406
00:23:31 --> 00:23:35
And this is, again,
supposed to be realistic.

407
00:23:35 --> 00:23:40
When I was three years old and
my brother's friend Ralph

408
00:23:40 --> 00:23:43
was seven, I watched him
throwing darts a lot.

409
00:23:43 --> 00:23:54
And I would say that for Ralph,
so for Ralph, at age seven,

410
00:23:54 --> 00:23:57
anyway, later on he got
a little better at it.

411
00:23:57 --> 00:24:00
But Ralph at age seven, the
probability that he hit

412
00:24:00 --> 00:24:08
the target was about 1/2.
 

413
00:24:08 --> 00:24:09
Right?
 

414
00:24:09 --> 00:24:12
So he hit the target
about half the time.

415
00:24:12 --> 00:24:15
And the other times, there was
cement on the walls of the

416
00:24:15 --> 00:24:17
basement, it wasn't that bad.
 

417
00:24:17 --> 00:24:19
Just bounced off.
 

418
00:24:19 --> 00:24:21
That also meant that the
points got a little

419
00:24:21 --> 00:24:22
blunter as time went on.
 

420
00:24:22 --> 00:24:26
So it was a little less
dangerous when they hit.

421
00:24:26 --> 00:24:26
Alright.
 

422
00:24:26 --> 00:24:33
So now, so here's the extra
assumption that I want to make.

423
00:24:33 --> 00:24:41
So a is going to be the
radius of the target.

424
00:24:41 --> 00:24:45
Now, the other realistic
assumption that I want to

425
00:24:45 --> 00:24:49
make is where this little
kid would be standing.

426
00:24:49 --> 00:24:52
And now, here, I want to get
very specific and just do the

427
00:24:52 --> 00:24:54
competition in one case.
 

428
00:24:54 --> 00:25:00
We're going to imagine
the target is here.

429
00:25:00 --> 00:25:02
And the kid is standing,
say, between, so we'll

430
00:25:02 --> 00:25:04
just do a section of this.
 

431
00:25:04 --> 00:25:13
This is between 3
o'clock and 5 o'clock.

432
00:25:13 --> 00:25:15
There's more of him, but
it's lower down and

433
00:25:15 --> 00:25:17
maybe negligible here.
 

434
00:25:17 --> 00:25:20
So this section is the
part, the chunk that

435
00:25:20 --> 00:25:23
we want to see about.
 

436
00:25:23 --> 00:25:31
And this is a, and then
this distance here is 2a.

437
00:25:31 --> 00:25:35
And then the longest
distance here is 3a.

438
00:25:35 --> 00:25:39
So the longest distance is 3a.
 

439
00:25:39 --> 00:25:45
So in other words, what I'm
saying is that the probability,

440
00:25:45 --> 00:25:58
if you're standing too close,
the chance Ralph hits younger

441
00:25:58 --> 00:26:06
brother is about 1/6, right?
 

442
00:26:06 --> 00:26:12
Because 2/12 = 1/6.
 

443
00:26:12 --> 00:26:22
1/6 of the probability that
we're between a and 3a.

444
00:26:22 --> 00:26:29
That's the number that
we're looking for.

445
00:26:29 --> 00:26:29
Another question.
 

446
00:26:29 --> 00:26:33
STUDENT: [INAUDIBLE]
 

447
00:26:33 --> 00:26:36
PROFESSOR: The 2/12 came from
the fact that we assumed.

448
00:26:36 --> 00:26:39
So we made a very, very
bold assumption here.

449
00:26:39 --> 00:26:42
We assumed that this human
being, who is actually

450
00:26:42 --> 00:26:46
standing, the floor
is about down here.

451
00:26:46 --> 00:26:48
Maybe he wasn't that tall.
 

452
00:26:48 --> 00:26:51
But anyway, so he's really
a little bigger than this.

453
00:26:51 --> 00:26:54
That the part of him that was
close to the target covered

454
00:26:54 --> 00:27:00
about this section here,
between radius 2a and 3a.

455
00:27:00 --> 00:27:03
As you'll see, actually from
the computation, because the

456
00:27:03 --> 00:27:08
likelihood drops off pretty
quickly, whatever of him was

457
00:27:08 --> 00:27:11
standing outside there wouldn't
have mattered anyway.

458
00:27:11 --> 00:27:13
So we're just worried about
the part that's closest

459
00:27:13 --> 00:27:19
to the target here.
 

460
00:27:19 --> 00:27:19
STUDENT: [INAUDIBLE]
 

461
00:27:19 --> 00:27:20
PROFESSOR: Why is it out of 12?
 

462
00:27:20 --> 00:27:22
Because I made it a clock.
 

463
00:27:22 --> 00:27:25
And I made it from 3 o'clock
to 5 o'clock, so it's 2 of

464
00:27:25 --> 00:27:28
the 12 hours of a clock.
 

465
00:27:28 --> 00:27:31
It's just a way of me
making a section that

466
00:27:31 --> 00:27:36
you can visibly see.
 

467
00:27:36 --> 00:27:40
So now, so here's what
we're trying to calculate.

468
00:27:40 --> 00:27:43
And in order to figure
this out, I need one

469
00:27:43 --> 00:27:47
more item. here.
 

470
00:27:47 --> 00:27:49
So maybe I'll leave myself
a little bit of room.

471
00:27:49 --> 00:27:52
I have to figure out
something about what our

472
00:27:52 --> 00:27:53
information gave us.
 

473
00:27:53 --> 00:27:59
Which is that the
probability, sorry, this

474
00:27:59 --> 00:28:01
probability was 1/2.
 

475
00:28:01 --> 00:28:03
So let's remember what this is.
 

476
00:28:03 --> 00:28:09
This is going to be e ^
- 0 ^2 - e ^ - a ^2.

477
00:28:09 --> 00:28:11
That's what's this
probability is.

478
00:28:11 --> 00:28:16
And that's equal to 1/2.
 

479
00:28:16 --> 00:28:21
So that means that 1
- e ^ - a ^2 = 1/2.

480
00:28:21 --> 00:28:26
Which means that e
^ - a ^2 = 1/2.

481
00:28:26 --> 00:28:29
I'm not going to calculate a,
this is the part about the

482
00:28:29 --> 00:28:32
information about a
that I want to use.

483
00:28:32 --> 00:28:33
That's what I'll use.
 

484
00:28:33 --> 00:28:36
And now I'm going to calculate
this other probability here.

485
00:28:36 --> 00:28:41
So the probability right
up there is this.

486
00:28:41 --> 00:28:50
And that's going to be the same
as e ^ - (2a)^2 - e ^ - (3a)^2.

487
00:28:51 --> 00:28:54
That's what we calculated.
 

488
00:28:54 --> 00:28:59
And now I want to use some of
the arithmetic of exponents.

489
00:28:59 --> 00:29:01
This is (e ^ - a ^2)^4.
 

490
00:29:01 --> 00:29:05
 


491
00:29:05 --> 00:29:10
Because it's really (2a)^2 = 4.
 

492
00:29:10 --> 00:29:15
The quantity is 4a^2, and then
I bring that exponent out. -

493
00:29:15 --> 00:29:25
(e ^ - a ^2) ^ 9 that's 3 ^2.
 

494
00:29:25 --> 00:29:31
And so this comes out to
be (1/2) ^ 4 - (1/2) ^9.

495
00:29:31 --> 00:29:36
 


496
00:29:36 --> 00:29:39
Which is approximately 1/16.
 

497
00:29:39 --> 00:29:40
This is negligible.
 

498
00:29:40 --> 00:29:41
This part here.
 

499
00:29:41 --> 00:29:43
And this is actually why these
tails, as you go out to

500
00:29:43 --> 00:29:48
infinity, don't really
matter that much.

501
00:29:48 --> 00:29:49
So this is a much
smaller number.

502
00:29:49 --> 00:29:53
So the probability of
the whole band is 1/16.

503
00:29:53 --> 00:29:55
And now I can answer
the question up here.

504
00:29:55 --> 00:30:01
This is approximately
1/6 * 1/16.

505
00:30:01 --> 00:30:07
Which is about
1/100, or about 1%.

506
00:30:07 --> 00:30:14
So if I stood there for 100
attempts, then my chances of

507
00:30:14 --> 00:30:18
getting hit were pretty high.
 

508
00:30:18 --> 00:30:27
So that's the computation.
 

509
00:30:27 --> 00:30:34
That's a typical example of
a problem in probability.

510
00:30:34 --> 00:30:37
And let me just make one
more connection with

511
00:30:37 --> 00:30:42
what we did before.
 

512
00:30:42 --> 00:30:46
This is connected to weighted
averages or integrals

513
00:30:46 --> 00:30:47
over weights.
 

514
00:30:47 --> 00:30:58
But the weight that's involved
in this problem was w ( r) =,

515
00:30:58 --> 00:31:00
so let's just look at what
happened in all of

516
00:31:00 --> 00:31:01
those integrals.
 

517
00:31:01 --> 00:31:04
What happened in all the
integrals was, we had

518
00:31:04 --> 00:31:08
this factor here.
 

519
00:31:08 --> 00:31:09
2 pi r.
 

520
00:31:09 --> 00:31:23
And if I include this c, it was
really 2 pi c r e ^ - r ^2.

521
00:31:23 --> 00:31:27
This was the weight
that we were using.

522
00:31:27 --> 00:31:30
The relative
importance of things.

523
00:31:30 --> 00:31:33
Now, this is not the
same as e ^ - r ^2 that

524
00:31:33 --> 00:31:34
we started out with.
 

525
00:31:34 --> 00:31:38
Because this is the
one-dimensional histogram.

526
00:31:38 --> 00:31:40
And that has to do with the
method of shells that gave us

527
00:31:40 --> 00:31:43
that extra factor of r here.
 

528
00:31:43 --> 00:31:46
So that also connects with the
question at the beginning.

529
00:31:46 --> 00:31:49
Which had to do with this
paradox, and it looks like

530
00:31:49 --> 00:31:51
these places in the middle
are the most likely.

531
00:31:51 --> 00:31:54
But that's the plot
of e ^ - r ^2.

532
00:31:54 --> 00:31:58
If you actually look at this
plot here, you see that as r

533
00:31:58 --> 00:32:00
goes to 0, it's going to 0.
 

534
00:32:00 --> 00:32:03
This is a different graph here.
 

535
00:32:03 --> 00:32:07
And actually, so this is
what's happening really.

536
00:32:07 --> 00:32:10
In terms of how likely it is
that you'll get within a

537
00:32:10 --> 00:32:18
certain distance of the
center of the target.

538
00:32:18 --> 00:32:20
Again, it's not the area under
that curve that we're doing.

539
00:32:20 --> 00:32:27
It's that volume of revolution.
 

540
00:32:27 --> 00:32:29
We're going to change
subjects now.

541
00:32:29 --> 00:32:30
OK, one more question.
 

542
00:32:30 --> 00:32:32
Yes.
 

543
00:32:32 --> 00:32:32
STUDENT: [INAUDIBLE]
 

544
00:32:32 --> 00:32:35
PROFESSOR: Yeah, that's
supposed to be the

545
00:32:35 --> 00:32:36
graph of w (r).
 

546
00:32:36 --> 00:32:47
STUDENT: [INAUDIBLE]
 

547
00:32:47 --> 00:32:49
PROFESSOR: Well, so, the
question is, wouldn't

548
00:32:49 --> 00:32:50
the importance of the
center be greatest?

549
00:32:50 --> 00:32:53
It's a question of which
variable you're using.

550
00:32:53 --> 00:32:57
According according to
pure radius, it's not.

551
00:32:57 --> 00:32:59
It turns out that there are
some bands in radius which are

552
00:32:59 --> 00:33:06
more important, more likely
for hits than others.

553
00:33:06 --> 00:33:08
It really has to do with the
fact that the center, or the

554
00:33:08 --> 00:33:10
core, of the target
is really tiny.

555
00:33:10 --> 00:33:11
So it's harder to hit.
 

556
00:33:11 --> 00:33:16
Whereas a whole band around the
outside has a lot more area.

557
00:33:16 --> 00:33:18
Many, many ways to
hit that band.

558
00:33:18 --> 00:33:19
So it's a much larger area.
 

559
00:33:19 --> 00:33:24
So there's a competition there
between those two things.

560
00:33:24 --> 00:33:25
So we're going to move on.
 

561
00:33:25 --> 00:33:28
And I want to talk now
about a different kind

562
00:33:28 --> 00:33:30
of weighted average.
 

563
00:33:30 --> 00:33:36
These weighted averages are
going to be much simpler.

564
00:33:36 --> 00:34:01
And they come up in what's
called numerical integration.

565
00:34:01 --> 00:34:04
There are many, many methods
of integrating numerically.

566
00:34:04 --> 00:34:06
And they're important because
many, many integrals

567
00:34:06 --> 00:34:08
don't have formulas.
 

568
00:34:08 --> 00:34:13
And so you have to compute them
with a calculator or a machine.

569
00:34:13 --> 00:34:24
So the first type that we've
already done are Riemann sums.

570
00:34:24 --> 00:34:28
They turned out to be
incredibly inefficient.

571
00:34:28 --> 00:34:30
They're lousy.
 

572
00:34:30 --> 00:34:32
The next rule that I'm
going to describe is

573
00:34:32 --> 00:34:33
a little improvement.
 

574
00:34:33 --> 00:34:38
It's called the
trapezoidal rule.

575
00:34:38 --> 00:34:41
And this one is much
more reasonable then

576
00:34:41 --> 00:34:43
the Riemann sum.
 

577
00:34:43 --> 00:34:49
Unfortunately, it's actually
also pretty lousy.

578
00:34:49 --> 00:34:53
There's another rule which is
just a slightly trickier rule.

579
00:34:53 --> 00:34:56
And it's actually
amazing that it exists.

580
00:34:56 --> 00:35:01
And it's called Simpson's Rule.
 

581
00:35:01 --> 00:35:06
And this one is
actually pretty good.

582
00:35:06 --> 00:35:07
It's clever.
 

583
00:35:07 --> 00:35:12
So let's get started
with these.

584
00:35:12 --> 00:35:15
And the way I'll get started
is by reminding you what

585
00:35:15 --> 00:35:16
the Riemann sum is.
 

586
00:35:16 --> 00:35:18
So this is a good review,
because you need to know

587
00:35:18 --> 00:35:19
all three of these.
 

588
00:35:19 --> 00:35:26
And you're going to want to see
them all laid out in parallel.

589
00:35:26 --> 00:35:31
So here's the setup.
 

590
00:35:31 --> 00:35:33
OK, so here we go.
 

591
00:35:33 --> 00:35:36
We have our graph, we have
our function it starts out

592
00:35:36 --> 00:35:38
at a, it ends up at b.
 

593
00:35:38 --> 00:35:40
Maybe goes on, but we're
only paying attention

594
00:35:40 --> 00:35:41
to this interval.
 

595
00:35:41 --> 00:35:44
This is a function y = f(x).
 

596
00:35:44 --> 00:35:48
And we split it up when
we do Riemann's sum.

597
00:35:48 --> 00:35:50
So this is 1, this
is Riemann sums.

598
00:35:50 --> 00:35:54
We start out with a, which is
a point we call x0, and then

599
00:35:54 --> 00:35:57
we go a certain distance.
 

600
00:35:57 --> 00:35:58
We go all the way over to b.
 

601
00:35:58 --> 00:36:04
And we subdivide this thing
with these delta x's.

602
00:36:04 --> 00:36:08
Which are the step sizes.
 

603
00:36:08 --> 00:36:12
So we have these little
steps of size delta x.

604
00:36:12 --> 00:36:16
Corresponding to these x
values, we have y values.

605
00:36:16 --> 00:36:21
y0 = f(x0), that's
the point above a.

606
00:36:21 --> 00:36:26
Then y1 = f ( x1), that's
the point above x1.

607
00:36:26 --> 00:36:33
And so forth, all the way
up to yn, which = f (xn).

608
00:36:33 --> 00:36:36
In order to figure out the
area, you really need to know

609
00:36:36 --> 00:36:37
something about the function.
 

610
00:36:37 --> 00:36:39
You need to be able
to evaluate it.

611
00:36:39 --> 00:36:40
So that's what we've done.
 

612
00:36:40 --> 00:36:43
We've evaluated here
at n + 1 points.

613
00:36:43 --> 00:36:46
Enumerated 0 through n.
 

614
00:36:46 --> 00:36:49
And those are the numbers out
of which we're going to get

615
00:36:49 --> 00:36:53
all of our approximations
to the integral.

616
00:36:53 --> 00:36:56
So somehow we want
average these numbers.

617
00:36:56 --> 00:37:01
So here's our goal.
 

618
00:37:01 --> 00:37:13
Our goal is to average, or
add, I'm using average

619
00:37:13 --> 00:37:14
very loosely here.
 

620
00:37:14 --> 00:37:18
But I was going to say
add up these numbers to

621
00:37:18 --> 00:37:25
get an approximation.
 

622
00:37:25 --> 00:37:30
To average or add the y's.
 

623
00:37:30 --> 00:37:37
To get an approximation
to the integral.

624
00:37:37 --> 00:37:43
Which we know is the
area under the curve.

625
00:37:43 --> 00:37:49
So here's what the
Riemann sum is.

626
00:37:49 --> 00:37:51
It's the following thing.
 

627
00:37:51 --> 00:37:56
You take (y0 + y1 + ...
up to yn - 1) And you

628
00:37:56 --> 00:37:57
multiply by delta x.
 

629
00:37:57 --> 00:38:00
That's it.
 

630
00:38:00 --> 00:38:07
Now, this one is the one
with left endpoints.

631
00:38:07 --> 00:38:09
So the left-hand sum.
 

632
00:38:09 --> 00:38:11
There's also a right one.
 

633
00:38:11 --> 00:38:13
Which is, if you start
at the right-hand ends.

634
00:38:13 --> 00:38:17
And that will go from y1 to yn.
 

635
00:38:17 --> 00:38:22
OK, so this one is
the right-hand.

636
00:38:22 --> 00:38:25
Right Riemann sum.
 

637
00:38:25 --> 00:38:33
Those are the two
that we did before.

638
00:38:33 --> 00:38:35
Now I'm going to describe
to you the next two.

639
00:38:35 --> 00:38:42
They have a similar
pattern to them.

640
00:38:42 --> 00:38:51
And the one with trapezoids
requires a picture.

641
00:38:51 --> 00:38:53
Here's a shape.
 

642
00:38:53 --> 00:39:00
And here's a bunch of values.
 

643
00:39:00 --> 00:39:08
And we're trying to estimate
the size of these chunks.

644
00:39:08 --> 00:39:11
And now, instead of doing
something stupid, which is to

645
00:39:11 --> 00:39:14
draw horizontal lines in
rectangles, we're going to do

646
00:39:14 --> 00:39:16
something slightly more clever.
 

647
00:39:16 --> 00:39:21
Which is to draw straight
lines that are diagonal.

648
00:39:21 --> 00:39:23
You see that many of them
actually coincide probably

649
00:39:23 --> 00:39:25
pretty closely with
what I drew there.

650
00:39:25 --> 00:39:29
Although if they're curved,
they miss by a little bit.

651
00:39:29 --> 00:39:32
So this is called the
trapezoidal rule.

652
00:39:32 --> 00:39:34
Because if you pick one of
these shapes, say, this is

653
00:39:34 --> 00:39:40
y2 and this is y3, if you
pick one of these shapes,

654
00:39:40 --> 00:39:43
this height here is y2.
 

655
00:39:43 --> 00:39:47
And this height is y3, and
this base is delta x.

656
00:39:47 --> 00:39:52
This is a trapezoid.
 

657
00:39:52 --> 00:40:01
So this being a trapezoid,
I can figure out its area.

658
00:40:01 --> 00:40:02
And what do I get?
 

659
00:40:02 --> 00:40:12
I get the base times
the average height.

660
00:40:12 --> 00:40:15
If you think about if, you work
out what happens when you do

661
00:40:15 --> 00:40:18
something with a straight line
on top, like that, you'll

662
00:40:18 --> 00:40:21
get this average.
 

663
00:40:21 --> 00:40:31
So this is the average
height of the trapezoid.

664
00:40:31 --> 00:40:34
And now I want to add up.
 

665
00:40:34 --> 00:40:41
I want to add them all up
to get my formula for

666
00:40:41 --> 00:40:45
the trapezoidal rule.
 

667
00:40:45 --> 00:40:46
So what do I do?
 

668
00:40:46 --> 00:40:50
I have delta x times
the first one.

669
00:40:50 --> 00:40:53
Which is y0 + y1 / 2.
 

670
00:40:53 --> 00:40:55
That's the first trapezoid.
 

671
00:40:55 --> 00:40:59
The next one is y1 + y2 / 2.
 

672
00:40:59 --> 00:41:01
And this keeps on going.
 

673
00:41:01 --> 00:41:07
And at the end, I have
yn - 2 + yn - 1 / 2.

674
00:41:07 --> 00:41:14
And then last of all, I
have yn - 1 + yn / 2.

675
00:41:14 --> 00:41:16
That's a very long
formula here.

676
00:41:16 --> 00:41:21
We're going to simplify it
quite a bit in just a second.

677
00:41:21 --> 00:41:22
What's this equal to?
 

678
00:41:22 --> 00:41:28
Well, notice that I get y0
/ 2 to start out with.

679
00:41:28 --> 00:41:32
And now, y1 got
mentioned twice.

680
00:41:32 --> 00:41:35
Each time with a factor of 1/2.
 

681
00:41:35 --> 00:41:39
So we get a whole y1 in here.
 

682
00:41:39 --> 00:41:41
And the same thing is
going to be true of

683
00:41:41 --> 00:41:43
all the middle terms.
 

684
00:41:43 --> 00:41:48
You're going to get y2 and
all the way up to yn - 1.

685
00:41:48 --> 00:41:53
But then, the last one is
unmatched. yn is only

686
00:41:53 --> 00:41:59
1/2, only counts 1/2.
 

687
00:41:59 --> 00:42:17
So here is what's known
as the trapezoidal rule.

688
00:42:17 --> 00:42:22
Now, I'd like to compare it
for you to the Riemann sums,

689
00:42:22 --> 00:42:25
which are sitting just
to the left here.

690
00:42:25 --> 00:42:28
Here's the left one, and
here's the right one.

691
00:42:28 --> 00:42:32
If you take the average of the
left and the right, that is,

692
00:42:32 --> 00:42:36
1/2 of this + 1/2 of that,
there's an overlap.

693
00:42:36 --> 00:42:39
The y1 through yn things
are listed in both.

694
00:42:39 --> 00:42:42
But the y0 only gets counted
1/2 and the yn only

695
00:42:42 --> 00:42:44
gets counted 1/2.
 

696
00:42:44 --> 00:42:47
So what this is, is this is the
symmetric compromise between

697
00:42:47 --> 00:42:49
the two Riemann sums.
 

698
00:42:49 --> 00:42:56
This is actually equal to
the left Riemann sum + the

699
00:42:56 --> 00:43:02
right Riemann sum / 2.
 

700
00:43:02 --> 00:43:10
It's the average of them.
 

701
00:43:10 --> 00:43:13
Now, this would be great and it
does look like it's closer.

702
00:43:13 --> 00:43:16
But actually it's not as
impressive as it looks.

703
00:43:16 --> 00:43:21
If you actually do it
in practice, it's

704
00:43:21 --> 00:43:23
not very efficient.
 

705
00:43:23 --> 00:43:25
Although it's way better
than a Riemann sum, it's

706
00:43:25 --> 00:43:27
still not good enough.
 

707
00:43:27 --> 00:43:37
So now I need to describe
to you the fancier rule.

708
00:43:37 --> 00:43:52
Which is known as
Simpson's Rule.

709
00:43:52 --> 00:44:00
And so, this is, if you
like, 3, Method 3.

710
00:44:00 --> 00:44:09
The idea is again to divide
things into chunks.

711
00:44:09 --> 00:44:16
But now it always
needs n to be even.

712
00:44:16 --> 00:44:19
In other words, we're going
to deal not with just one

713
00:44:19 --> 00:44:24
box, we're going to deal
with pairs of boxes.

714
00:44:24 --> 00:44:28
Here's delta x, and
here's delta x again.

715
00:44:28 --> 00:44:34
And we're going to study the
area of this piece here.

716
00:44:34 --> 00:44:39
So let me focus
just on that part.

717
00:44:39 --> 00:44:50
Let's reproduce it over here.
 

718
00:44:50 --> 00:44:52
And here's the delta
x, here's delta x.

719
00:44:52 --> 00:44:54
And of course there
are various heights.

720
00:44:54 --> 00:44:57
This starts out at y0,
this is y2, and this

721
00:44:57 --> 00:45:03
middle segment is y1.
 

722
00:45:03 --> 00:45:06
Now, the approximating
curve that we're going

723
00:45:06 --> 00:45:14
to use is a parabola.
 

724
00:45:14 --> 00:45:16
That is, we're going to
fit a parabola through

725
00:45:16 --> 00:45:25
these three points.
 

726
00:45:25 --> 00:45:28
And then we're going to use
that as the approximating area.

727
00:45:28 --> 00:45:32
Now, it doesn't look like,
this looks like it misses.

728
00:45:32 --> 00:45:34
But actually, most functions
mostly wiggle either

729
00:45:34 --> 00:45:35
one way or the other.
 

730
00:45:35 --> 00:45:36
They don't switch.
 

731
00:45:36 --> 00:45:38
They don't have
inflection points.

732
00:45:38 --> 00:45:41
So, this is a lousy,
at this scale.

733
00:45:41 --> 00:45:43
But when we get to a smaller
scale, this becomes

734
00:45:43 --> 00:45:45
really fantastic.
 

735
00:45:45 --> 00:45:47
As an approximation.
 

736
00:45:47 --> 00:45:52
Now, I need to tell you
what the arithmetic is.

737
00:45:52 --> 00:45:57
And in order to save time,
it's on your problem set

738
00:45:57 --> 00:46:00
what the actual formula is.
 

739
00:46:00 --> 00:46:05
But I'm going to tell you
how to think about it.

740
00:46:05 --> 00:46:09
I want you to think
about it as follows.

741
00:46:09 --> 00:46:24
So the area under the parabola
is going to be a base times

742
00:46:24 --> 00:46:30
some kind of average height.
 

743
00:46:30 --> 00:46:33
And the base here,
you can already see.

744
00:46:33 --> 00:46:36
It's 2 delta x.
 

745
00:46:36 --> 00:46:39
The base is 2 delta x.
 

746
00:46:39 --> 00:46:40
Now, the average
height is weird.

747
00:46:40 --> 00:46:43
You have to work out what it is
for a parabola, depending on

748
00:46:43 --> 00:46:46
those three numbers.
y0, y1, and y2.

749
00:46:46 --> 00:46:48
And it turns out to be
the following formula.

750
00:46:48 --> 00:46:51
It has to be an average,
but it's an interesting

751
00:46:51 --> 00:46:51
weighted average.
 

752
00:46:51 --> 00:46:54
So this was the
punchline, if you like.

753
00:46:54 --> 00:46:55
Is that there were such
things as interesting

754
00:46:55 --> 00:46:56
weighted averages.
 

755
00:46:56 --> 00:46:58
This one's very simple, it
just involves three numbers.

756
00:46:58 --> 00:46:59
But it's still interesting.
 

757
00:46:59 --> 00:47:00
It's the following.
 

758
00:47:00 --> 00:47:07
It turns out to be
y0 + 4 y1 + y2 / 6.

759
00:47:07 --> 00:47:08
Why divided by 6?
 

760
00:47:08 --> 00:47:10
Well, it's supposed
to be an average.

761
00:47:10 --> 00:47:13
So the total is 1 + 4
+ 1 of these things.

762
00:47:13 --> 00:47:16
And 6 is in the denominator.
 

763
00:47:16 --> 00:47:19
So it emphasizes the middle
more than the sides.

764
00:47:19 --> 00:47:23
And that's what happens
with a parabola.

765
00:47:23 --> 00:47:30
So this is a computation
which is on your homework.

766
00:47:30 --> 00:47:33
And now we can put this
together for the full

767
00:47:33 --> 00:47:36
Simpson's Rule formula.
 

768
00:47:36 --> 00:47:51
Which I'll put up over here.
 

769
00:47:51 --> 00:47:56
We have here 2 delta x,
and we divide by 6.

770
00:47:56 --> 00:48:02
And then we have y0 +
4 y1 + y2 / 6 +...

771
00:48:03 --> 00:48:05
that's the first chunk.
 

772
00:48:05 --> 00:48:08
Now, the second chunk,
maybe I'll just put it in

773
00:48:08 --> 00:48:11
here, starts this is x2.
 

774
00:48:11 --> 00:48:14
This is x0.
 

775
00:48:14 --> 00:48:15
And it goes all the way to x4.
 

776
00:48:15 --> 00:48:18
So x2, x3, x4.
 

777
00:48:18 --> 00:48:21
So the next one involves
the indices 2, 3 and 4.

778
00:48:21 --> 00:48:31
So this is y2 + 4 y3 + y4 / 6.
 

779
00:48:31 --> 00:48:34
Oh, oh, oh, oh, no.
 

780
00:48:34 --> 00:48:35
I think I'll get
rid of these 6's.

781
00:48:35 --> 00:48:37
I have too many 6's.
 

782
00:48:37 --> 00:48:39
Alright.
 

783
00:48:39 --> 00:48:41
Let's get rid of them here.
 

784
00:48:41 --> 00:48:41
Let's take them out.
 

785
00:48:41 --> 00:48:44
Put them out here.
 

786
00:48:44 --> 00:48:47
Thank you.
 

787
00:48:47 --> 00:48:53
All the way to the end, which
is y(n - 2) + 2 y(n - 1) -

788
00:48:53 --> 00:48:57
sorry, + 4 y(n - 1) + yn.
 

789
00:48:57 --> 00:49:01
I was about to divide by
6, but you saved me.

790
00:49:01 --> 00:49:04
So here are all the chunks.
 

791
00:49:04 --> 00:49:06
Now, what does this
pattern come out to be?

792
00:49:06 --> 00:49:10
This comes out to
be the following.

793
00:49:10 --> 00:49:16
This is 1, 4, 1, added to
1, 4, 1 added to 1, 4, 1.

794
00:49:16 --> 00:49:17
You add them up.
 

795
00:49:17 --> 00:49:22
You 1, 4, and then there's a
repeat, so you get a 2 and a

796
00:49:22 --> 00:49:24
4, and a 2 and a 4 and a 1.
 

797
00:49:24 --> 00:49:27
So the pattern is that
it starts out with

798
00:49:27 --> 00:49:29
1's on the far ends.
 

799
00:49:29 --> 00:49:30
And then 4's next in.
 

800
00:49:30 --> 00:49:35
And then it alternates
2's and 4's in between.

801
00:49:35 --> 00:49:44
So the full pattern of
Simpson's Rule is delta x /

802
00:49:44 --> 00:49:47
3, I have now succeeded in
canceling this 2 with this

803
00:49:47 --> 00:49:50
6 and getting out
that factor of 2.

804
00:49:50 --> 00:50:01
And then here I have y0 + 4 y1
+ 2 y2 + 4 y3 + ... it keeps on

805
00:50:01 --> 00:50:03
going and keeps on going
and keeps on going.

806
00:50:03 --> 00:50:10
And in the end it's 2 y(n
- 2) + 4 y(n - 1) + yn.

807
00:50:10 --> 00:50:14
So again, 1 and a 4 to start.
 

808
00:50:14 --> 00:50:16
1 and a 4 and a 1 to end.
 

809
00:50:16 --> 00:50:20
And then alternating 2's and
4's in the middle And this

810
00:50:20 --> 00:50:25
weird weighted average
is way better.

811
00:50:25 --> 00:50:28
As I will show you next time.
 

812
00:50:28 --> 00:50:29