What is a differential equation

WEBVTT
Kind: captions
Language: en

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Welcome to this first video, and
actually the first video

00:00:04.170 --> 00:00:06.800
in the playlist on differential
equations.

00:00:06.800 --> 00:00:09.810
I know I touched on this before
when we did harmonic

00:00:09.810 --> 00:00:11.170
motion, and I think I
might have touched

00:00:11.170 --> 00:00:12.450
on it in other subjects.

00:00:12.450 --> 00:00:16.400
But now, because of your
request, we'll do a whole

00:00:16.400 --> 00:00:17.180
playlist on this.

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And that's a fairly useful
thing, because differential

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equations is something that
shows up in a whole set of

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different fields.

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I've been requested by someone
who's starting an economics

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PhD program to do this; I've
been requested by some people

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who are going into physics, some
people who are going into

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engineering.

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So it's a widely applicable
area of study.

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So let's just get started,
before I keep going off on

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useless stuff.

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So the differential equations.

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So the first question is: what
is a differential equation?

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You know what an equation is.

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What is a differential
equation?

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Well, a differential equation
is an equation that involves

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an unknown function and
its derivatives.

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So what do I mean by that?

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Well, let's say that I said that
y prime plus y is equal

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to x plus 3.

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Here, the unknown
function is y.

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We could have written it as y of
x, or we could have written

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this as dy dx, the derivative
of y with respect to x plus

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this unknown function y
is equal to plus 3.

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We also could have written f
prime of x plus f of x is

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equal to plus 3.

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All of these would have been
valid ways of writing this

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exact same differential
equation.

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And what's interesting here,
and how this is a departure

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from what we've learned before
about just regular equations

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is that-- let me write down
a regular equation just to

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remind you what they
look like.

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So a regular equation, if we had
one variable, would look

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something like this.

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I don't know, x squared plus
the cosine of x is equal to

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the square root of x.

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I just made that up.

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Here, the solution is a number,
or sometimes it's a

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set of numbers.

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Sometimes there's more
than one, right?

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If you have a polynomial, you
could have more than one

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values of x that satisfy
this equation.

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Here, for a differential
equation, the

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solution is a function.

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Our goal is to figure out what
function of x, and here I

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wrote f of x explicitly, but
what function of x explicitly

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satisfies this relationship
or this equation.

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So let me show you what
I mean by that.

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And I have my differential
equations book from college,

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so I'm going to use
that as we go.

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So let's say that-- I'm
just writing now.

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See, they have this as a
differential equation.

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And I'm not going to show you
necessarily how to solve them

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just yet, because we have to
learn some tricks first. But I

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think a good place to start is
just so you understand what a

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differential equation is, so
you don't get confused with

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the traditional equation.

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So, they have this differential
[? derivative. ?]

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y prime prime.

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So the second derivative of y
with respect to x, plus 2

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times the first derivative of y
with respect to x, minus 3 y

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is equal to 0.

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And they give us the solutions
here, and what they want us to

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do is show that these
are solutions.

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And I think this is a good
place to just at least

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understand what a differential
equation is, and what its

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solution means.

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So they say y1 of x is equal
to e to the minus 3x.

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So they claim that this
is a solution of this

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differential equation.

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So let me show to you
that this is.

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Well, if this is soon. y1,
what's y-- well, let

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me just write y1.

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What's y1 prime?

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What's the derivative of this?

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Well, just do the chain rule.

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The derivative of the whole
function, with respect to this

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part of it, is just
e to the minus 3x.

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And then you take the derivative
of the inside.

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So that's just the derivative
of the outside, e

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to the minus 3x.

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And the derivative of the
inside is minus 3.

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And the second derivative of
y1 is equal to-- we'll just

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take the derivative of this, and
that's just equal to plus

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9-- minus 3 times minus
3-- e to the minus 3x.

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Now, let's verify that if
we substitute y1 and its

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derivatives back into this
differential equation, that it

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holds true.

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So y prime prime, that's this.

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So we get nine e to the minus
3x, plus 2y prime.

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Plus 2 times y prime.

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Well, this is y prime.

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So 2 times minus 3 e to the
minus 3x plus-- oh sorry,

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minus-- 3 times y.

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Well, y is this.

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So minus 3 times e
to the minus 3x.

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Well, what does that equal?

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We get 9 e to the minus 3x,
minus 6 e to the minus 3x,

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minus 3 e to the minus 3x.

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Well, what does that equal?

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We have 9 of something
minus 6 of

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something minus 3 of something.

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So that just equals 0.

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It doesn't matter
of 0 whatever.

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So that equals 0.

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So we verified that for this
function, for y1 is equal to e

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to the minus 3x, it satisfies
this differential equation.

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Now there's something
interesting here, and you've

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kind of touched on this with
regular equations, is that

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this might not be the
only solution.

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In fact we'll learn, in maybe a
video or two, that often the

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solution is not just
a function.

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It could be a class of functions
where usually

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they're all kind of the same
function, but you have a

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difference of constants.

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But I'll show you that
in a second.

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But here, they actually show
us that there's another

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solution, that this will
actually work with, we could

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try the equation y2 of x
is equal to, well, just

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simple e to the x.

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And we could verify
that, right?

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What's the first and second
derivatives of e to the x?

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Well, they're just e to the x.

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So the second derivative of y2
is just e to the x plus 2

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times the first derivative
is what?

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Well the first derivative of e
to the x is still e to the x,

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2 e to the x, minus 3
times a function.

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Minus 3e to the x.

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Well, 1 plus 2 minus 3, well
that equals 0 again.

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So this was also a solution to
this differential equation.

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Now before we go on, in the next
one I'll show you some

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fairly straightforward
differential

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equations to solve.

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I think it's a good time now,
now that you hopefully have a

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grasp of what a differential
equation is, and what its

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solution is.

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And its solution isn't a number,
its solution is a

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function, or a set
of functions,

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or a class of functions.

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It's a good time to just
go over a little bit of

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terminology.

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So there's two big
classifications.

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Well actually, there's a first
big one, ordinary and partial

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differential equations.

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I think you might have already
guessed what that means.

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An ordinary differential
equation is what I wrote down.

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It's one variable with respect
to another variable, or one

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function with respect you to,
say, x and its derivatives.

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Partial differential equations
we'll get into later.

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That's more complicated.

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That's when a function
can be a function of

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more than one variable.

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And you can have the derivative
with respect to x,

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and y, and z.

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We won't worry about
that right now.

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If your functions and their
derivatives are a function of

00:07:36.570 --> 00:07:38.630
only one variable, then we're
dealing with an ordinary

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differential equation.

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That's what this playlist
will deal with, ordinary

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differential equations.

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Now within ordinary differential
equations,

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there's two ways of
classifying, and

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they kind of overlap.

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You have your order, so what
is the order of my

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differential equation?

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And then you have this notion
of whether it is linear or

00:08:05.820 --> 00:08:08.100
non-linear.

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And I think the best way to
figure this out is just to

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write down examples.

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So let me write down one.

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And I'm getting this
from my college

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differential equations book.

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x squared times the second
derivative of y with respect

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to x, plus x times the first
derivative of y with respect

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to x, plus 2y is equal
to sine of x.

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So the first question here
is: what is the order?

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All the order is is the highest
derivative that exists

00:08:47.270 --> 00:08:48.600
in your equation.

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The highest derivative
of the function

00:08:50.360 --> 00:08:51.150
under question, right?

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The solution of this is going to
be a y of x, that satisfies

00:08:55.650 --> 00:08:56.880
this equation.

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And the order is the highest
derivative of that function.

00:08:59.750 --> 00:09:04.080
Well, the highest derivative
here is the second derivative.

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So this has order 2.

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Or as you could call this,
a second order ordinary

00:09:11.960 --> 00:09:13.360
differential equation.

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Now the second thing we have to
figure out: is this linear

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or is this a non-linear
differential equation?

00:09:18.640 --> 00:09:23.600
So a differential equation is
linear if all of the functions

00:09:23.600 --> 00:09:28.480
and its derivatives are
essentially, well for lack of

00:09:28.480 --> 00:09:29.110
a better word, linear.

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What do I mean by that?

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I mean you don't have a y
squared, or you don't have a

00:09:32.510 --> 00:09:36.750
dy over dx squared, or you
don't have a y times the

00:09:36.750 --> 00:09:38.080
second derivative of y.

00:09:38.080 --> 00:09:43.100
So this example I just wrote
here, this is a second order

00:09:43.100 --> 00:09:46.530
linear equation, because you
have the second derivative,

00:09:46.530 --> 00:09:49.580
the first derivative, and y, but
they're not multiplied by

00:09:49.580 --> 00:09:50.910
the function or the
derivatives.

00:09:50.910 --> 00:09:59.600
Now if this equation were-- if
I rewrote it as x squared d,

00:09:59.600 --> 00:10:06.260
the second derivative of y with
respect to x squared, is

00:10:06.260 --> 00:10:11.470
equal to sine of x, and let's
say I were to square this.

00:10:11.470 --> 00:10:13.660
Now, all of the sudden,
I have a non-linear

00:10:13.660 --> 00:10:14.690
differential equation.

00:10:14.690 --> 00:10:15.860
This is non-linear.

00:10:15.860 --> 00:10:17.030
This is linear.

00:10:17.030 --> 00:10:21.190
Because I squared, I multiplied
the second

00:10:21.190 --> 00:10:25.410
derivative of y with respect--
I multiplied it times itself.

00:10:25.410 --> 00:10:27.740
Another example of a non-linear
equation is if I

00:10:27.740 --> 00:10:35.230
wrote y times the second
derivative of y with respect

00:10:35.230 --> 00:10:38.390
to x is equal to sine of x.

00:10:38.390 --> 00:10:41.800
This is also non-linear, because
I multiplied the

00:10:41.800 --> 00:10:44.140
function times its second
derivative.

00:10:44.140 --> 00:10:46.340
Notice here, I did multiply
stuff times the second

00:10:46.340 --> 00:10:49.010
derivative, but it was the
independent variable x that I

00:10:49.010 --> 00:10:50.200
multiplied.

00:10:50.200 --> 00:10:52.690
But anyway, I've run out of
time, and hopefully that gives

00:10:52.690 --> 00:10:54.690
you a good at least
survey of what a

00:10:54.690 --> 00:10:56.270
differential equation is.

00:10:56.270 --> 00:10:59.840
In the next video, we'll start
actually solving them.

00:10:59.840 --> 00:11:01.090
See


Triple integrals 3

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.280
Let's now do another triple
integral, and in this one I

00:00:03.280 --> 00:00:05.300
won't actually evaluate
the triple integral.

00:00:05.300 --> 00:00:08.860
But what we'll do is we'll
define the triple integral.

00:00:08.860 --> 00:00:10.310
We're going to something
similar that we did in the

00:00:10.310 --> 00:00:12.500
second video where we
figured out the mass

00:00:12.500 --> 00:00:13.670
using a density function.

00:00:13.670 --> 00:00:16.380
But what I want to do in this
video is show you how to set

00:00:16.380 --> 00:00:18.010
the boundaries when the
figure is a little

00:00:18.010 --> 00:00:19.010
bit more complicated.

00:00:19.010 --> 00:00:21.350
And if we have time we'll try
to do it where we change

00:00:21.350 --> 00:00:22.470
the order of integration.

00:00:22.470 --> 00:00:26.090
So let's say I have the surface
-- let me just make something

00:00:26.090 --> 00:00:39.340
up -- 2x plus 3z plus
y is equal to 6.

00:00:39.340 --> 00:00:41.516
Let's draw that surface.

00:00:41.516 --> 00:00:43.070
It looks something like this.

00:00:43.070 --> 00:00:46.720
This will be my x-axis.

00:00:46.720 --> 00:00:49.580
This is going to be my z-axis.

00:00:49.580 --> 00:00:50.860
This is going to be my y-axis.

00:00:53.910 --> 00:00:56.694
Draw them out.

00:00:56.694 --> 00:00:59.990
x, y and z.

00:00:59.990 --> 00:01:03.600
And I care about the surface in
the kind of positive octant,

00:01:03.600 --> 00:01:04.970
right, because when you're
dealing with three-dimensionals

00:01:04.970 --> 00:01:06.840
we have, instead of four
quadrants we have

00:01:06.840 --> 00:01:07.380
eight octants.

00:01:07.380 --> 00:01:10.280
But we want the octant where
all x, y and z is positive,

00:01:10.280 --> 00:01:11.780
which is the one I drew here.

00:01:11.780 --> 00:01:14.740
So let's see, let me draw some
-- what is the x-intercept?

00:01:14.740 --> 00:01:17.440
When y and z are 0,
so we'll write here,

00:01:17.440 --> 00:01:18.860
that's the x-intercept.

00:01:18.860 --> 00:01:21.880
2x is equal to 6, so
x is equal to 3.

00:01:21.880 --> 00:01:25.070
So 1, 2, 3.

00:01:25.070 --> 00:01:26.910
So that's the x-intercept.

00:01:26.910 --> 00:01:30.880
The y-intercept when x and
z are 0 are on the y-axis,

00:01:30.880 --> 00:01:32.470
so y will be equal to 6.

00:01:32.470 --> 00:01:41.000
so we have 1, 2, 3, 4, 5,
6 is the y-intercept.

00:01:41.000 --> 00:01:44.160
Then finally the z-intercept
when x and y are 0

00:01:44.160 --> 00:01:45.140
we're on the z-axis.

00:01:45.140 --> 00:01:47.620
3z will be equal to 6.

00:01:47.620 --> 00:01:51.240
So z is equal to 1, 2.

00:01:51.240 --> 00:01:55.880
So the figure that I care about
will look something like

00:01:55.880 --> 00:02:02.730
this -- it's be this
inclined surface.

00:02:02.730 --> 00:02:06.150
It will look
something like that.

00:02:06.150 --> 00:02:08.740
In this positive octant.

00:02:08.740 --> 00:02:11.070
So this is the surface
defined by this function.

00:02:11.070 --> 00:02:13.545
Let's say that I care about
the volume, and I'm going

00:02:13.545 --> 00:02:15.540
to make it a little
bit more complicated.

00:02:15.540 --> 00:02:17.320
We could say oh, well let's find the volume between the

00:02:17.320 --> 00:02:19.170
surface and the xy plane.

00:02:19.170 --> 00:02:20.710
But I'm going to make it a
little bit more complicated.

00:02:20.710 --> 00:02:24.440
Let's say the volume above
this surface, and the

00:02:24.440 --> 00:02:28.530
surface z is equal to 2.

00:02:28.530 --> 00:02:32.830
So the volume we care
about is going to look

00:02:32.830 --> 00:02:33.690
something like this.

00:02:33.690 --> 00:02:37.345
Let me see if I can
pull off drawing this.

00:02:37.345 --> 00:02:46.130
If we go up 2 here -- let me
draw the top in a different

00:02:46.130 --> 00:02:49.430
color, let me draw
the top in green.

00:02:49.430 --> 00:02:53.530
So this is along the zy plane.

00:02:53.530 --> 00:02:56.270
And then the other
edge is going to look

00:02:56.270 --> 00:02:58.636
something like this.

00:02:58.636 --> 00:03:02.110
Let me make sure I can draw it
-- this is the hardest part.

00:03:02.110 --> 00:03:06.350
We'll go up 2 here, and this
is along the zx plane.

00:03:08.970 --> 00:03:10.930
And we'd have another line
connecting these two.

00:03:13.740 --> 00:03:16.150
So this green triangle,
this is part of the

00:03:16.150 --> 00:03:18.180
plane z is equal to 2.

00:03:21.700 --> 00:03:25.450
The volume we care about is the
volume between this top green

00:03:25.450 --> 00:03:29.540
plane and this tilted plane
defined by 2x plus 3z

00:03:29.540 --> 00:03:31.100
plus y is equal to 6.

00:03:31.100 --> 00:03:32.530
So this area in between.

00:03:32.530 --> 00:03:34.780
Let me see if I can make
it a little bit clearer.

00:03:34.780 --> 00:03:37.750
Because the visualization, as I
say, is often the hardest part.

00:03:37.750 --> 00:03:42.590
So we'd have kind of a front
wall here, and then the back

00:03:42.590 --> 00:03:46.660
wall would be this wall back
here, and then there'd

00:03:46.660 --> 00:03:50.020
be another wall here.

00:03:50.020 --> 00:03:53.250
And then the base of it, the
base I'll do in magenta

00:03:53.250 --> 00:03:55.330
will be this plane.

00:03:55.330 --> 00:03:59.140
So the base is that plane
-- that's the bottom part.

00:03:59.140 --> 00:04:01.190
Anyway, I don't know if I
should have made it that messy

00:04:01.190 --> 00:04:04.120
because we're going to have to
draw dv's and d volumes on it.

00:04:04.120 --> 00:04:06.830
But anyway, let's try our best.

00:04:06.830 --> 00:04:10.130
So, if we're going to figure
out the volume -- and actually,

00:04:10.130 --> 00:04:12.230
since we're doing a triple
integral and we want to show

00:04:12.230 --> 00:04:14.630
that we have to do the triple
integral, instead of doing a

00:04:14.630 --> 00:04:18.730
volume, let's do the mass of
something of variable density.

00:04:18.730 --> 00:04:22.700
So let's say the density in
this volume that we care about,

00:04:22.700 --> 00:04:27.240
the density function is a
function of x, y and z.

00:04:27.240 --> 00:04:28.220
It can be anything.

00:04:28.220 --> 00:04:29.770
That's not the point of what
I'm trying to teach here.

00:04:29.770 --> 00:04:30.860
But I'll just define something.

00:04:30.860 --> 00:04:33.820
Let's say it's x squared yz.

00:04:33.820 --> 00:04:36.080
Our focus is really just
to set up the integrals.

00:04:36.080 --> 00:04:39.130
So the first thing I like to do
is I visualize -- what we're

00:04:39.130 --> 00:04:42.080
going to do is we're going to
set up a little cube in the

00:04:42.080 --> 00:04:43.410
volume under consideration.

00:04:43.410 --> 00:04:46.750
So if I had a -- let me do it
in a bold color so that you can

00:04:46.750 --> 00:04:51.565
see it -- so if I have a cube
-- maybe I'll do it in brown,

00:04:51.565 --> 00:04:53.400
it's not as bold but it's
different enough from

00:04:53.400 --> 00:04:54.590
the other colors.

00:04:54.590 --> 00:05:00.810
So if I had a little cube
here in the volume under

00:05:00.810 --> 00:05:06.480
consideration, that's a little
cube -- you consider that dv.

00:05:06.480 --> 00:05:09.840
The volume of that cube is kind
of a volume differential.

00:05:09.840 --> 00:05:13.470
And that is equal to dx --
no, sorry, this is dy.

00:05:13.470 --> 00:05:16.760
Let me do this in yellow,
or green even better.

00:05:16.760 --> 00:05:19.820
So dy, which is this.

00:05:19.820 --> 00:05:24.370
dy times dx, dx times dz.

00:05:24.370 --> 00:05:27.420
That's the volume of
that little cube.

00:05:27.420 --> 00:05:29.520
And if we wanted to know the
mass of that cube, we would

00:05:29.520 --> 00:05:33.650
multiply the density function
at that point times this dv.

00:05:33.650 --> 00:05:36.810
So the mass, you could call
it d -- I don't know, dm.

00:05:36.810 --> 00:05:41.270
The mass differential is going
to be equal to that times that.

00:05:41.270 --> 00:05:46.230
So x squared y z times this.

00:05:46.230 --> 00:05:48.610
dy, dx, and dz.

00:05:48.610 --> 00:05:51.850
And we normally switch this
order around, depending on what

00:05:51.850 --> 00:05:53.650
we're going to integrate with
respect to first so we

00:05:53.650 --> 00:05:54.910
don't get confused.

00:05:54.910 --> 00:05:56.210
So let's try to do this.

00:05:56.210 --> 00:05:57.430
Let's try to set
up this integral.

00:05:57.430 --> 00:05:58.900
So let's do it traditionally.

00:05:58.900 --> 00:06:01.250
The last couple of triple
integrals we did we integrated

00:06:01.250 --> 00:06:03.770
with respect to z first.

00:06:03.770 --> 00:06:04.580
So let's do that.

00:06:04.580 --> 00:06:07.155
So we're going to integrate
with respect to z first. so

00:06:07.155 --> 00:06:11.750
we're going to take this cube
and we're going to sum up all

00:06:11.750 --> 00:06:13.280
of the cubes in the z-axis.

00:06:13.280 --> 00:06:16.130
So going up and
down first, right?

00:06:16.130 --> 00:06:18.500
So if we do that, what
is the bottom boundary?

00:06:23.430 --> 00:06:25.890
So when you sum up up and down,
these cubes are going to

00:06:25.890 --> 00:06:27.460
turn to columns, right?

00:06:27.460 --> 00:06:30.620
So what is the bottom of the
column, the bottom bound?

00:06:30.620 --> 00:06:31.700
What's the surface?

00:06:31.700 --> 00:06:33.970
It's the surface
defined right here.

00:06:33.970 --> 00:06:37.740
So, if we want that bottom
bound defined in terms of

00:06:37.740 --> 00:06:39.800
z, we just have to solve
this in terms of z.

00:06:39.800 --> 00:06:40.880
So let's subtract.

00:06:40.880 --> 00:06:41.950
So what do we get.

00:06:41.950 --> 00:06:46.210
If we want this defined in
terms of z, we get 3z is

00:06:46.210 --> 00:06:51.160
equal to 6 minus 2x minus y.

00:06:51.160 --> 00:06:57.900
Or z is equal 2 minus
2/3x minus y over 3.

00:06:57.900 --> 00:06:59.380
This is the same thing as that.

00:06:59.380 --> 00:07:02.385
But when we're talking about
z, explicitly defining a

00:07:02.385 --> 00:07:05.070
z, this is how we get it,
algebraically manipulated.

00:07:05.070 --> 00:07:07.740
So the bottom boundary -- and
you can visualize it, right?

00:07:07.740 --> 00:07:09.670
The bottom of these columns
are going to go up and down.

00:07:09.670 --> 00:07:11.490
We're going to add up all
the columns in up and

00:07:11.490 --> 00:07:12.200
down direction, right?

00:07:12.200 --> 00:07:13.680
You can imagine summing them.

00:07:13.680 --> 00:07:18.020
The bottom boundary is
going to be this surface.

00:07:18.020 --> 00:07:25.860
z is equal to 2 minus
2/3x minus y over 3.

00:07:25.860 --> 00:07:26.840
And then what's
the upper bound?

00:07:26.840 --> 00:07:29.440
Well, the top of the column
is going to be this green

00:07:29.440 --> 00:07:31.610
plane, and what did we
say the green plane was?

00:07:31.610 --> 00:07:33.110
It was z is equal to 2.

00:07:36.130 --> 00:07:37.890
That's this plane, this
surface right here.

00:07:37.890 --> 00:07:40.970
Z is equal to 2.

00:07:40.970 --> 00:07:43.690
And, of course, what is the
volume of that column?

00:07:43.690 --> 00:07:49.500
Well, it's going to be the
density function, x squared yz

00:07:49.500 --> 00:07:52.820
times the volume differential,
but we're integrating

00:07:52.820 --> 00:07:54.005
with respect to z first.

00:07:54.005 --> 00:07:57.020
Let me write dz there.

00:07:57.020 --> 00:08:00.950
I don't know, let's say we want
to integrate with respect to --

00:08:00.950 --> 00:08:02.450
I don't know, we want to
integrate with respect

00:08:02.450 --> 00:08:03.540
to x for next.

00:08:03.540 --> 00:08:05.030
In the last couple of
videos, I integrated

00:08:05.030 --> 00:08:05.930
with respect to y next.

00:08:05.930 --> 00:08:08.880
So let's do x just to show you
it really doesn't matter.

00:08:08.880 --> 00:08:11.650
So we're going to integrate
with respect to x.

00:08:11.650 --> 00:08:13.760
So, now we have these
columns, right?

00:08:13.760 --> 00:08:15.920
When we integrate with respect
to z, we get the volume of each

00:08:15.920 --> 00:08:19.060
of these columns wher the
top boundary is that plane.

00:08:19.060 --> 00:08:22.220
Let's see if I can
draw it decently.

00:08:22.220 --> 00:08:24.570
The top boundary is that plane.

00:08:24.570 --> 00:08:27.810
The bottom boundary
is this surface.

00:08:27.810 --> 00:08:29.560
Now we want to integrate
with respect to x.

00:08:29.560 --> 00:08:32.720
So we're going to add
up all of the dx's.

00:08:32.720 --> 00:08:36.620
So what is the bottom
boundary for the x's?

00:08:36.620 --> 00:08:41.620
Well, this surface is defined
all the way to -- the volume

00:08:41.620 --> 00:08:44.590
under question is defined all
the way until x is equal to 0.

00:08:44.590 --> 00:08:48.110
And if you get confused, and
it's not that difficult to get

00:08:48.110 --> 00:08:50.260
confused when you're imagining
these three-dimensional things,

00:08:50.260 --> 00:08:52.910
say you know what, we already
integrated with respect to z.

00:08:52.910 --> 00:08:55.530
The two variables I
have left are x and y.

00:08:55.530 --> 00:08:59.530
Let me draw the projection of
our volume onto the xy plane,

00:08:59.530 --> 00:09:01.300
and what does that look like?

00:09:01.300 --> 00:09:02.350
So I will do that.

00:09:02.350 --> 00:09:04.900
Because that actually does
help simplify things.

00:09:04.900 --> 00:09:07.550
So if we twist it, if we take
this y and flip it out like

00:09:07.550 --> 00:09:11.570
that, and x like that we'll get
in kind of the traditional way

00:09:11.570 --> 00:09:14.664
that we learned when we
first learned algebra.

00:09:14.664 --> 00:09:15.600
The xy-axis.

00:09:18.745 --> 00:09:21.900
So this is x, this is y.

00:09:21.900 --> 00:09:24.510
And this point is what?

00:09:24.510 --> 00:09:25.300
Or this point?

00:09:25.300 --> 00:09:25.660
What is that?

00:09:25.660 --> 00:09:28.060
That's x is equal to 3.

00:09:28.060 --> 00:09:30.710
So it's 1, 2, 3.

00:09:30.710 --> 00:09:31.600
That's x is equal to 3.

00:09:31.600 --> 00:09:33.930
And this point right here
is y is equal to 6.

00:09:33.930 --> 00:09:39.200
So 1, 2, 3, 4, 5, 6.

00:09:39.200 --> 00:09:43.900
So on the xy-axis, kind of the
domain -- you can view it that

00:09:43.900 --> 00:09:46.360
-- looks something like that.

00:09:46.360 --> 00:09:50.540
So one way to think about it is
we've figured out if these

00:09:50.540 --> 00:09:55.000
columns -- we've integrated
up/down or along the z-axis.

00:09:55.000 --> 00:09:58.790
But when you view it looking
straight down onto it, you're

00:09:58.790 --> 00:10:01.350
looking on the xy plane, each
of our columns are going to

00:10:01.350 --> 00:10:06.100
look like this where the
column's going to pop out out

00:10:06.100 --> 00:10:07.320
of your screen in
the z direction.

00:10:07.320 --> 00:10:10.760
But the base of each column is
going to dx like that, and

00:10:10.760 --> 00:10:13.040
then dy up and down, right?

00:10:13.040 --> 00:10:15.800
So we decided to integrate
with respect to x next.

00:10:15.800 --> 00:10:18.520
So we're going to add up each
of those columns in the

00:10:18.520 --> 00:10:20.860
x direction, in the
horizontal direction.

00:10:20.860 --> 00:10:23.140
So the question was what
is the bottom boundary?

00:10:23.140 --> 00:10:24.880
What is the lower bound
in the x direction?

00:10:24.880 --> 00:10:26.680
Well, it's x is equal to 0.

00:10:26.680 --> 00:10:30.400
If there was a line here, then
it would be that line probably

00:10:30.400 --> 00:10:34.740
as a function of y, or
definitely as a function of y.

00:10:34.740 --> 00:10:38.800
So our bottom bound here
is x is equal to 0.

00:10:38.800 --> 00:10:39.620
What's our top bound?

00:10:39.620 --> 00:10:41.340
I realize I'm already pushing.

00:10:41.340 --> 00:10:43.490
Well, our top bound is this
relation, but it has to

00:10:43.490 --> 00:10:46.100
be in terms of x, right?

00:10:46.100 --> 00:10:46.980
So what is this relation.

00:10:46.980 --> 00:10:50.230
So, you could view it as kind
of saying well, if z is equal

00:10:50.230 --> 00:10:51.840
to 0, what is this line?

00:10:51.840 --> 00:10:52.770
What is this line right here?

00:10:52.770 --> 00:10:53.910
So z is equal to 0.

00:10:53.910 --> 00:10:57.390
We have 2x plus y
is equal to 6.

00:10:57.390 --> 00:10:59.590
We want the relationship
in terms of x.

00:10:59.590 --> 00:11:04.130
So we get 2x is equal to 6
minus y where x is equal

00:11:04.130 --> 00:11:06.220
to 3 minus y over 2.

00:11:11.130 --> 00:11:13.500
And then finally we're going to
integrate with respect to y.

00:11:13.500 --> 00:11:16.130
And this is the easy part.

00:11:16.130 --> 00:11:20.530
So we've integrated up and
down to get a column.

00:11:20.530 --> 00:11:23.150
These are the bases of the
column, so we've integrated

00:11:23.150 --> 00:11:24.000
in the x direction.

00:11:24.000 --> 00:11:26.480
Now we just have to go up and
down with respect to y, or in

00:11:26.480 --> 00:11:28.350
the xy plane with respect to y.

00:11:28.350 --> 00:11:29.970
So what is the y
bottom boundary?

00:11:29.970 --> 00:11:31.060
Well, it's 0.

00:11:31.060 --> 00:11:32.650
y is equal to 0.

00:11:32.650 --> 00:11:35.660
And the top boundary
is y is equal to 6.

00:11:35.660 --> 00:11:36.540
And there you have it.

00:11:36.540 --> 00:11:38.730
We have set up the integral
and now it's just a matter

00:11:38.730 --> 00:11:41.590
of chugging through
it mechanically.

00:11:41.590 --> 00:11:43.120
But I've run out of time
and I don't want this

00:11:43.120 --> 00:11:44.660
video to get rejected.

00:11:44.660 --> 00:11:45.870
So I'll leave you there.

00:11:45.870 --> 00:11:47.760
See you in the next video.


Triple integrals 2

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:03.820
In the last video, we had this
rectangle, and we used a triple

00:00:03.820 --> 00:00:05.170
integral to figure
out it's volume.

00:00:05.170 --> 00:00:08.000
And I know you were probably
thinking, well, I could have

00:00:08.000 --> 00:00:12.100
just used my basic geometry
to multiply the height times

00:00:12.100 --> 00:00:12.940
the width times the depth.

00:00:12.940 --> 00:00:15.720
And that's true because this
was a constant-valued function.

00:00:15.720 --> 00:00:18.320
And then even once we
evaluated, once we integrated

00:00:18.320 --> 00:00:20.630
with respect to z, we ended up
with a double integral, which

00:00:20.630 --> 00:00:23.840
is exactly what you would have
done in the last several videos

00:00:23.840 --> 00:00:26.580
when we just learned the
volume under a surface.

00:00:26.580 --> 00:00:28.560
But then we added a twist
at the end of the video.

00:00:28.560 --> 00:00:33.000
We said, fine, you could have
figured out the volume within

00:00:33.000 --> 00:00:38.160
this rectangular domain, I
guess, very straightforward

00:00:38.160 --> 00:00:39.000
using things you already knew.

00:00:39.000 --> 00:00:42.080
But what if our goal is not
to figure out the volume?

00:00:42.080 --> 00:00:46.790
Our goal was to figure out the
mass of this volume, and even

00:00:46.790 --> 00:00:50.240
more, the material that we're
taking the volume of-- whether

00:00:50.240 --> 00:00:53.060
it's a volume of gas or a
volume of some solid-- that

00:00:53.060 --> 00:00:55.050
its density is not constant.

00:00:55.050 --> 00:00:58.080
So now the mass becomes
kind of-- I don't know--

00:00:58.080 --> 00:00:59.550
interesting to calculate.

00:00:59.550 --> 00:01:03.740
And so, what we defined, we
defined a density function.

00:01:03.740 --> 00:01:07.770
And rho, this p looking thing
with a curvy bottom-- that

00:01:07.770 --> 00:01:09.855
gives us the density
at any given point.

00:01:09.855 --> 00:01:11.370
And at the end of the
last video we said,

00:01:11.370 --> 00:01:12.830
well, what is mass?

00:01:12.830 --> 00:01:16.050
Mass is just density
times volume.

00:01:16.050 --> 00:01:16.750
You could view it another way.

00:01:16.750 --> 00:01:21.170
Density is the same thing
as mass divided by volume.

00:01:21.170 --> 00:01:26.630
So the mass around a very, very
small point, and we called that

00:01:26.630 --> 00:01:29.750
d mass, the differential of the
mass, is going equal the

00:01:29.750 --> 00:01:33.450
density at that point, or the
rough density at exactly that

00:01:33.450 --> 00:01:36.790
point, times the volume
differential around that point,

00:01:36.790 --> 00:01:40.100
times the volume of this
little small cube.

00:01:40.100 --> 00:01:43.140
And then, as we saw it on the
last video, if you're using

00:01:43.140 --> 00:01:46.240
rectangular coordinates, this
volume differential could just

00:01:46.240 --> 00:01:50.390
be the x distance times the y
distance times the z distance.

00:01:50.390 --> 00:01:55.690
So, the density was that our
density function is defined

00:01:55.690 --> 00:01:57.730
to be x, y, and z, and we
wanted to figure out the

00:01:57.730 --> 00:02:01.560
mass of this volume.

00:02:01.560 --> 00:02:04.140
And let's say that our x, y,
and z coordinates-- their

00:02:04.140 --> 00:02:05.990
values, let's say they're in
meters and let's say this

00:02:05.990 --> 00:02:09.340
density is in kilograms
per meter cubed.

00:02:09.340 --> 00:02:12.270
So our answer is going to be in
kilograms if that was the case.

00:02:12.270 --> 00:02:14.480
And those are kind of the
traditional Si units.

00:02:14.480 --> 00:02:21.210
So let's figure out the mass of
this variably dense volume.

00:02:21.210 --> 00:02:24.150
So all we do is we have the
same integral up here.

00:02:26.720 --> 00:02:29.860
So the differential of mass
is going to be this value,

00:02:29.860 --> 00:02:30.996
so let's write that down.

00:02:34.850 --> 00:02:38.920
It is x-- I want to make sure
I don't run out of space.

00:02:38.920 --> 00:02:43.390
xyz times-- and I'm
going to integrate with

00:02:43.390 --> 00:02:45.890
respect to dz first.

00:02:45.890 --> 00:02:47.910
But you could actually
switch the order.

00:02:47.910 --> 00:02:49.750
Maybe we'll do that
in the next video.

00:02:49.750 --> 00:02:55.810
We'll do dz first, then we'll
do dy, then we'll do dx.

00:03:00.120 --> 00:03:02.490
Once again, this is just
the mass at any small

00:03:02.490 --> 00:03:04.310
differential of volume.

00:03:04.310 --> 00:03:07.760
And if we integrate with z
first we said z goes from what?

00:03:07.760 --> 00:03:10.770
The boundaries on
z were 0 to 2.

00:03:14.050 --> 00:03:18.255
The boundaries on
y were 0 to 4.

00:03:21.110 --> 00:03:23.890
And the boundaries on
x, x went from 0 to 3.

00:03:26.750 --> 00:03:27.910
And how do we evaluate this?

00:03:27.910 --> 00:03:29.900
Well, what is the
antiderivative-- we're

00:03:29.900 --> 00:03:31.370
integrating with
respect to z first.

00:03:31.370 --> 00:03:35.660
So what's the antiderivative
of xyz with respect to z?

00:03:35.660 --> 00:03:37.080
Well, let's see.

00:03:37.080 --> 00:03:45.080
This is just a constant so
it'll be xyz squared over 2.

00:03:45.080 --> 00:03:46.040
Right?

00:03:46.040 --> 00:03:46.810
Yeah, that's right.

00:03:46.810 --> 00:03:52.690
And then we'll evaluate
that from 2 to 0.

00:03:52.690 --> 00:03:54.870
And so you get-- I know I'm
going to run out of space.

00:03:54.870 --> 00:03:59.420
So you're going to get
2 squared, which is 4,

00:03:59.420 --> 00:04:00.990
divided by 2, which is 2.

00:04:00.990 --> 00:04:05.460
So it's 2xy minus 0.

00:04:05.460 --> 00:04:09.070
So when you evaluate just this
first we'll get 2xy, and

00:04:09.070 --> 00:04:11.410
now you have the other
two integrals left.

00:04:11.410 --> 00:04:13.260
So I didn't write the
other two integrals down.

00:04:13.260 --> 00:04:13.820
Maybe I'll write it down.

00:04:13.820 --> 00:04:16.680
So then you're left
with two integrals.

00:04:16.680 --> 00:04:20.660
You're left with dy and dx.

00:04:20.660 --> 00:04:28.710
And y goes from 0 to 4
and x goes from 0 to 3.

00:04:28.710 --> 00:04:30.480
I'm definitely going
to run out of space.

00:04:30.480 --> 00:04:32.200
And now you take the
antiderivative of this

00:04:32.200 --> 00:04:34.110
with respect to y.

00:04:34.110 --> 00:04:36.640
So what's the antiderivative
of this with respect to y?

00:04:36.640 --> 00:04:40.240
Let me erase some stuff just
so I don't get too messy.

00:04:44.230 --> 00:04:46.040
I was given the very good
suggestion of making it

00:04:46.040 --> 00:04:48.340
scroll, but, unfortunately,
I didn't make it scroll

00:04:48.340 --> 00:04:50.090
enough this time.

00:04:50.090 --> 00:04:54.160
So I can delete this
stuff, I think.

00:04:54.160 --> 00:04:55.220
Oops, I deleted some of that.

00:04:55.220 --> 00:04:56.860
But you know what I deleted.

00:04:56.860 --> 00:04:58.290
OK, so let's take
the antiderivative

00:04:58.290 --> 00:04:59.290
with respect to y.

00:04:59.290 --> 00:05:02.640
I'll start it up here
where I have space.

00:05:02.640 --> 00:05:06.545
OK, so the antiderivative of
2xy with respect to y is y

00:05:06.545 --> 00:05:08.430
squared over 2, 2's cancel out.

00:05:08.430 --> 00:05:09.870
So you get xy squared.

00:05:13.100 --> 00:05:15.270
And y goes from 0 to 4.

00:05:15.270 --> 00:05:18.000
And then we still have the
outer integral to do.

00:05:18.000 --> 00:05:22.395
x goes from 0 to 3 dx.

00:05:22.395 --> 00:05:24.215
And when y is equal
to 4 you get 16x.

00:05:27.050 --> 00:05:29.050
And then when y is 0
the whole thing is 0.

00:05:29.050 --> 00:05:34.300
So you have 16x integrated
from 0 to 3 dx.

00:05:34.300 --> 00:05:36.210
And that is equal to what?

00:05:36.210 --> 00:05:39.215
8x squared.

00:05:39.215 --> 00:05:42.700
And you evaluate
it from 0 to 3.

00:05:42.700 --> 00:05:46.560
When it's 3, 8 times 9 is 72.

00:05:46.560 --> 00:05:49.040
And 0 times 8 is 0.

00:05:49.040 --> 00:05:51.810
So the mass of our figure-- the
volume we figured out last

00:05:51.810 --> 00:05:53.230
time was 24 meters cubed.

00:05:53.230 --> 00:05:55.160
I erased it, but if you
watched the last video

00:05:55.160 --> 00:05:56.210
that's what we learned.

00:05:56.210 --> 00:06:00.570
But it's mass is 72 kilograms.

00:06:00.570 --> 00:06:06.420
And we did that by integrating
this 3 variable density

00:06:06.420 --> 00:06:08.090
function-- this function
of 3 variables.

00:06:08.090 --> 00:06:10.230
Or in three-dimensions
you can view it as a

00:06:10.230 --> 00:06:11.440
scalar field, right?

00:06:11.440 --> 00:06:13.910
At any given point, there
is a value, but not

00:06:13.910 --> 00:06:14.420
really a direction.

00:06:14.420 --> 00:06:16.020
And that value is a density.

00:06:16.020 --> 00:06:20.540
But we integrated the scalar
field in this volume.

00:06:20.540 --> 00:06:22.650
So that's kind of the new
skill we learned with

00:06:22.650 --> 00:06:23.620
the triple integral.

00:06:23.620 --> 00:06:26.280
And in the next video I'll
show you how to set up more

00:06:26.280 --> 00:06:27.460
complicated triple integrals.

00:06:27.460 --> 00:06:29.820
But the real difficulty with
triple integrals is-- and I

00:06:29.820 --> 00:06:32.180
think you'll see that your
calculus teacher will often do

00:06:32.180 --> 00:06:34.630
this-- when you're doing triple
integrals, unless you have a

00:06:34.630 --> 00:06:38.290
very easy figure like this, the
evaluation-- if you actually

00:06:38.290 --> 00:06:41.500
wanted to analytically evaluate
a triple integral that has more

00:06:41.500 --> 00:06:44.910
complicated boundaries or more
complicated for example,

00:06:44.910 --> 00:06:46.280
a density function.

00:06:46.280 --> 00:06:48.850
The integral gets very
hairy, very fast.

00:06:48.850 --> 00:06:52.610
And it's often very difficult
or very time consuming to

00:06:52.610 --> 00:06:55.760
evaluate it analytically just
using your traditional

00:06:55.760 --> 00:06:56.270
calculus skills.

00:06:56.270 --> 00:06:59.790
So you'll see that on a lot of
calculus exams when they start

00:06:59.790 --> 00:07:02.500
doing the triple integral, they
just want you to set it up.

00:07:02.500 --> 00:07:05.520
They take your word for it that
you've done so many integrals

00:07:05.520 --> 00:07:07.490
so far that you could
take the antiderivative.

00:07:07.490 --> 00:07:09.820
And sometimes, if they really
want to give you something more

00:07:09.820 --> 00:07:12.530
difficult they'll just say,
well, switch the order.

00:07:12.530 --> 00:07:14.930
You know, this is the integral
when we're dealing with

00:07:14.930 --> 00:07:16.700
respect to z, then y, then x.

00:07:16.700 --> 00:07:18.510
We want you to rewrite
this integral when

00:07:18.510 --> 00:07:19.730
you switch the order.

00:07:19.730 --> 00:07:22.700
And we will do that
in the next video.

00:07:22.700 --> 00:07:24.270
See you soon.


Triple integrals 1

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.160
Let's say I wanted to find the
volume of a cube, where the

00:00:04.160 --> 00:00:07.150
values of the cube-- let's say
x is between-- x is greater

00:00:07.150 --> 00:00:10.350
than or equal to 0, is less
than or equal to,

00:00:10.350 --> 00:00:11.780
I don't know, 3.

00:00:11.780 --> 00:00:14.600
Let's say y is greater than
or equal to 0, and is

00:00:14.600 --> 00:00:17.000
less than or equal to 4.

00:00:17.000 --> 00:00:21.270
And then let's say that z is
greater than or equal to 0 and

00:00:21.270 --> 00:00:23.055
is less than or equal to 2.

00:00:23.055 --> 00:00:26.650
And I know, using basic
geometry you could figure out--

00:00:26.650 --> 00:00:30.370
you know, just multiply the
width times the height times

00:00:30.370 --> 00:00:31.340
the depth and you'd
have the volume.

00:00:31.340 --> 00:00:34.280
But I want to do this example,
just so that you get used to

00:00:34.280 --> 00:00:36.700
what a triple integral looks
like, how it relates to a

00:00:36.700 --> 00:00:39.180
double integral, and then later
in the next video we could do

00:00:39.180 --> 00:00:40.290
something slightly
more complicated.

00:00:40.290 --> 00:00:44.040
So let's just draw
that, this volume.

00:00:44.040 --> 00:00:51.780
So this is my x-axis, this is
my z-axis, this is the y.

00:00:54.330 --> 00:00:55.795
x, y, z.

00:00:59.600 --> 00:01:00.080
OK.

00:01:00.080 --> 00:01:01.910
So x is between 0 and 3.

00:01:01.910 --> 00:01:03.070
So that's x is equal to 0.

00:01:03.070 --> 00:01:09.120
This is x is equal to--
let's see, 1, 2, 3.

00:01:09.120 --> 00:01:10.570
y is between 0 and 4.

00:01:10.570 --> 00:01:13.180
1, 2, 3, 4.

00:01:13.180 --> 00:01:15.450
So the x-y plane will look
something like this.

00:01:15.450 --> 00:01:20.520
The kind of base of our cube
will look something like this.

00:01:20.520 --> 00:01:21.770
And then z is between 0 and 2.

00:01:21.770 --> 00:01:25.350
So 0 is the x-y plane,
and then 1, 2.

00:01:25.350 --> 00:01:27.130
So this would be the top part.

00:01:27.130 --> 00:01:30.600
And maybe I'll do that in a
slightly different color.

00:01:30.600 --> 00:01:34.520
So this is along the x-z axis.

00:01:34.520 --> 00:01:36.360
You'd have a boundary
here, and then it would

00:01:36.360 --> 00:01:38.316
come in like this.

00:01:38.316 --> 00:01:41.850
You have a boundary here,
come in like that.

00:01:41.850 --> 00:01:43.810
A boundary there.

00:01:43.810 --> 00:01:45.600
So we want to figure out
the volume of this cube.

00:01:45.600 --> 00:01:46.370
And you could do it.

00:01:46.370 --> 00:01:51.540
You could say, well, the depth
is 3, the base, the width is 4,

00:01:51.540 --> 00:01:53.920
so this area is 12
times the height.

00:01:53.920 --> 00:01:55.170
12 times 2 is 24.

00:01:55.170 --> 00:01:58.980
You could say it's 24
cubic units, whatever

00:01:58.980 --> 00:01:59.630
units we're doing.

00:01:59.630 --> 00:02:01.990
But let's do it as
a triple integral.

00:02:01.990 --> 00:02:03.640
So what does a triple
integral mean?

00:02:03.640 --> 00:02:07.110
Well, what we could do is we
could take the volume of a very

00:02:07.110 --> 00:02:10.670
small-- I don't want to say
area-- of a very small volume.

00:02:10.670 --> 00:02:14.770
So let's say I wanted to take
the volume of a small cube.

00:02:14.770 --> 00:02:17.810
Some place in this-- in the
volume under question.

00:02:17.810 --> 00:02:20.160
And it'll start to make more
sense, or it starts to become a

00:02:20.160 --> 00:02:22.860
lot more useful, when we have
variable boundaries and

00:02:22.860 --> 00:02:25.050
surfaces and curves
as boundaries.

00:02:25.050 --> 00:02:26.840
But let's say we want to
figure out the volume of this

00:02:26.840 --> 00:02:29.780
little, small cube here.

00:02:29.780 --> 00:02:30.590
That's my cube.

00:02:30.590 --> 00:02:33.630
It's some place in this larger
cube, this larger rectangle,

00:02:33.630 --> 00:02:35.460
cubic rectangle, whatever
you want to call it.

00:02:35.460 --> 00:02:36.540
So what's the volume
of that cube?

00:02:36.540 --> 00:02:38.930
Let's say that its width is dy.

00:02:42.320 --> 00:02:44.010
So that length
right there is dy.

00:02:44.010 --> 00:02:46.810
It's height is dx.

00:02:46.810 --> 00:02:49.660
Sorry, no, it's
height is dz, right?

00:02:49.660 --> 00:02:51.840
The way I drew it,
z is up and down.

00:02:51.840 --> 00:02:53.860
And it's depth is dx.

00:02:53.860 --> 00:02:55.940
This is dx.

00:02:55.940 --> 00:02:56.750
This is dz.

00:02:56.750 --> 00:02:57.720
This is dy.

00:02:57.720 --> 00:03:01.260
So you can say that a small
volume within this larger

00:03:01.260 --> 00:03:04.830
volume-- you could call that
dv, which is kind of the

00:03:04.830 --> 00:03:06.750
volume differential.

00:03:06.750 --> 00:03:10.290
And that would be equal to,
you could say, it's just

00:03:10.290 --> 00:03:13.990
the width times the
length times the height.

00:03:13.990 --> 00:03:15.950
dx times dy times dz.

00:03:15.950 --> 00:03:17.760
And you could switch the
orders of these, right?

00:03:17.760 --> 00:03:21.010
Because multiplication is
associative, and order

00:03:21.010 --> 00:03:22.920
doesn't matter and all that.

00:03:22.920 --> 00:03:24.540
But anyway, what can you
do with it in here?

00:03:24.540 --> 00:03:27.290
Well, we can take the integral.

00:03:27.290 --> 00:03:32.520
All integrals help us do is
help us take infinite sums of

00:03:32.520 --> 00:03:36.080
infinitely small distances,
like a dz or a dx or

00:03:36.080 --> 00:03:38.240
a dy, et cetera.

00:03:38.240 --> 00:03:41.620
So, what we could do is we
could take this cube and

00:03:41.620 --> 00:03:44.110
first, add it up in, let's
say, the z direction.

00:03:44.110 --> 00:03:48.330
So we could take that cube and
then add it along the up and

00:03:48.330 --> 00:03:51.230
down axis-- the z-axis--
so that we get the

00:03:51.230 --> 00:03:52.410
volume of a column.

00:03:52.410 --> 00:03:54.550
So what would that look like?

00:03:54.550 --> 00:03:56.930
Well, since we're going up and
down, we're adding-- we're

00:03:56.930 --> 00:04:00.670
taking the sum in
the z direction.

00:04:00.670 --> 00:04:02.610
We'd have an integral.

00:04:02.610 --> 00:04:04.655
And then what's the
lowest z value?

00:04:04.655 --> 00:04:08.310
Well, it's z is equal to 0.

00:04:08.310 --> 00:04:09.280
And what's the upper bound?

00:04:09.280 --> 00:04:12.070
Like if you were to just take--
keep adding these cubes, and

00:04:12.070 --> 00:04:14.190
keep going up, you'd run
into the upper bound.

00:04:14.190 --> 00:04:14.770
And what's the upper bound?

00:04:14.770 --> 00:04:16.100
It's z is equal to 2.

00:04:20.580 --> 00:04:25.010
And of course, you would
take the sum of these dv's.

00:04:25.010 --> 00:04:26.130
And I'll write dz first.

00:04:26.130 --> 00:04:28.170
Just so it reminds us
that we're going to

00:04:28.170 --> 00:04:30.430
take the integral with
respect to z first.

00:04:30.430 --> 00:04:32.010
And let's say we'll do y next.

00:04:32.010 --> 00:04:34.200
And then we'll do x.

00:04:34.200 --> 00:04:37.430
So this integral, this value,
as I've written it, will

00:04:37.430 --> 00:04:42.020
figure out the volume of a
column given any x and y.

00:04:42.020 --> 00:04:45.240
It'll be a function of x and y,
but since we're dealing with

00:04:45.240 --> 00:04:47.130
all constants here, it's
actually going to be

00:04:47.130 --> 00:04:48.600
a constant value.

00:04:48.600 --> 00:04:52.160
It'll be the constant value
of the volume of one

00:04:52.160 --> 00:04:53.890
of these columns.

00:04:53.890 --> 00:04:56.580
So essentially, it'll
be 2 times dy dx.

00:04:56.580 --> 00:04:59.330
Because the height of one
of these columns is 2,

00:04:59.330 --> 00:05:03.710
and then its with and
its depth is dy and dx.

00:05:03.710 --> 00:05:06.570
So then if we want to figure
out the entire volume-- what

00:05:06.570 --> 00:05:09.270
we did just now is we figured
out the height of a column.

00:05:09.270 --> 00:05:11.300
So then we could take those
columns and sum them

00:05:11.300 --> 00:05:13.730
in the y direction.

00:05:13.730 --> 00:05:15.710
So if we're summing in the y
direction, we could just take

00:05:15.710 --> 00:05:20.340
another integral of this
sum in the y direction.

00:05:20.340 --> 00:05:25.650
And y goes from 0 to what?
y goes from 0 to 4.

00:05:25.650 --> 00:05:27.180
I wrote this integral a
little bit too far to the

00:05:27.180 --> 00:05:28.300
left, it looks strange.

00:05:28.300 --> 00:05:31.000
But I think you get the idea.

00:05:31.000 --> 00:05:33.390
y is equal to 0, to
y is equal to 4.

00:05:33.390 --> 00:05:37.420
And then that'll give us the
volume of a sheet that is

00:05:37.420 --> 00:05:40.290
parallel to the zy plane.

00:05:40.290 --> 00:05:44.250
And then all we have left to do
is add up a bunch of those

00:05:44.250 --> 00:05:46.570
sheets in the x direction, and
we'll have the volume

00:05:46.570 --> 00:05:48.210
of our entire figure.

00:05:48.210 --> 00:05:50.190
So to add up those sheets,
we would have to sum

00:05:50.190 --> 00:05:51.750
in the x direction.

00:05:51.750 --> 00:05:57.060
And we'd go from x is equal
to 0, to x is equal to 3.

00:05:57.060 --> 00:05:58.660
And to evaluate this
is actually fairly

00:05:58.660 --> 00:05:59.690
straightforward.

00:05:59.690 --> 00:06:03.020
So, first we're taking the
integral with respect to z.

00:06:03.020 --> 00:06:05.090
Well, we don't have anything
written under here, but we

00:06:05.090 --> 00:06:06.740
can just assume that
there's a 1, right?

00:06:06.740 --> 00:06:10.160
Because dz times dy times
dx is the same thing as

00:06:10.160 --> 00:06:12.940
1 times dz times dy dx.

00:06:12.940 --> 00:06:15.500
So what's the value
of this integral?

00:06:15.500 --> 00:06:18.760
Well, the antiderivative
of 1 with respect to

00:06:18.760 --> 00:06:20.650
z is just z, right?

00:06:20.650 --> 00:06:22.700
Because the derivative
of z is 1.

00:06:22.700 --> 00:06:27.640
And you evaluate
that from 2 to 0.

00:06:27.640 --> 00:06:30.210
So then you're left with--
so it's 2 minus 0.

00:06:30.210 --> 00:06:31.580
So you're just left with 2.

00:06:31.580 --> 00:06:34.390
So you're left with 2, and you
take the integral of that from

00:06:34.390 --> 00:06:38.080
y is equal to 0, to y is equal
to 4 dy, and then

00:06:38.080 --> 00:06:40.060
you have the x.

00:06:40.060 --> 00:06:45.280
From x is equal to 0,
to x is equal to 3 dx.

00:06:45.280 --> 00:06:48.440
And notice, when we just took
the integral with respect to

00:06:48.440 --> 00:06:50.210
z, we ended up with
a double integral.

00:06:50.210 --> 00:06:52.830
And this double integral is the
exact integral we would have

00:06:52.830 --> 00:06:56.440
done in the previous videos on
the double integral, where you

00:06:56.440 --> 00:06:59.510
would have just said, well,
z is a function of x and y.

00:06:59.510 --> 00:07:01.880
So you could have written, you
know, z, is a function of x

00:07:01.880 --> 00:07:04.230
and y, is always equal to 2.

00:07:04.230 --> 00:07:05.180
It's a constant function.

00:07:05.180 --> 00:07:06.980
It's independent of x and y.

00:07:06.980 --> 00:07:09.210
But if you had defined z in
this way, and you wanted to

00:07:09.210 --> 00:07:11.985
figure out the volume under
this surface, where the surface

00:07:11.985 --> 00:07:15.370
is z is equal to 2-- you
know, this is a surface, is z

00:07:15.370 --> 00:07:17.580
is equal to 2-- we would
have ended up with this.

00:07:17.580 --> 00:07:19.130
So you see that what we're
doing with the triple

00:07:19.130 --> 00:07:21.030
integral, it's really,
really nothing different.

00:07:21.030 --> 00:07:22.060
And you might be wondering,
well, why are we

00:07:22.060 --> 00:07:22.840
doing it at all?

00:07:22.840 --> 00:07:25.730
And I'll show you
that in a second.

00:07:25.730 --> 00:07:28.320
But anyway, to evaluate
this, you could take the

00:07:28.320 --> 00:07:32.070
antiderivative of this with
respect to y, you get 2y-- let

00:07:32.070 --> 00:07:33.760
me scroll down a little bit.

00:07:33.760 --> 00:07:38.530
You get 2y evaluating
that at 4 and 0.

00:07:38.530 --> 00:07:41.150
And then, so you get 2 times 4.

00:07:41.150 --> 00:07:42.540
So it's 8 minus 0.

00:07:42.540 --> 00:07:46.070
And then you integrate
that from, with respect

00:07:46.070 --> 00:07:48.340
to x from 0 to 3.

00:07:48.340 --> 00:07:52.430
So that's 8x from 0 to 3.

00:07:52.430 --> 00:07:55.430
So that'll be equal to
24 four units cubed.

00:07:55.430 --> 00:07:59.780
So I know the obvious question
is, what is this good for?

00:07:59.780 --> 00:08:05.420
Well, when you have a kind
of a constant value within

00:08:05.420 --> 00:08:06.400
the volume, you're right.

00:08:06.400 --> 00:08:08.230
You could have just done
a double integral.

00:08:08.230 --> 00:08:11.610
But what if I were to tell you,
our goal is not to figure out

00:08:11.610 --> 00:08:13.670
the volume of this figure.

00:08:13.670 --> 00:08:16.550
Our goal is to figure out
the mass of this figure.

00:08:16.550 --> 00:08:21.660
And even more, this volume--
this area of space or

00:08:21.660 --> 00:08:23.670
whatever-- its mass
is not uniform.

00:08:23.670 --> 00:08:28.190
If its mass was uniform, you
could just multiply its uniform

00:08:28.190 --> 00:08:31.240
density times its volume,
and you'd get its mass.

00:08:31.240 --> 00:08:33.040
But let's say the
density changes.

00:08:33.040 --> 00:08:36.340
It could be a volume of some
gas or it could be even some

00:08:36.340 --> 00:08:39.070
material with different
compounds in it.

00:08:39.070 --> 00:08:42.370
So let's say that its density
is a variable function

00:08:42.370 --> 00:08:43.240
of x, y, and z.

00:08:43.240 --> 00:08:47.650
So let's say that the density--
this row, this thing that looks

00:08:47.650 --> 00:08:50.720
like a p is what you normally
use in physics for density-- so

00:08:50.720 --> 00:08:54.390
its density is a function
of x, y, and z.

00:08:54.390 --> 00:08:55.710
Let's-- just to make it
simple-- let's make

00:08:55.710 --> 00:08:59.840
it x times y times z.

00:08:59.840 --> 00:09:06.020
If we wanted to figure out the
mass of any small volume, it

00:09:06.020 --> 00:09:08.440
would be that volume times
the density, right?

00:09:08.440 --> 00:09:12.190
Because density-- the units of
density are like kilograms

00:09:12.190 --> 00:09:13.590
per meter cubed.

00:09:13.590 --> 00:09:16.400
So if you multiply it times
meter cubed, you get kilograms.

00:09:16.400 --> 00:09:20.260
So we could say that the mass--
well, I'll make up notation, d

00:09:20.260 --> 00:09:23.730
mass-- this isn't a function.

00:09:23.730 --> 00:09:25.230
Well, I don't want to write it
in parentheses, because it

00:09:25.230 --> 00:09:26.230
makes it look like a function.

00:09:26.230 --> 00:09:30.490
So, a very differential mass,
or a very small mass, is going

00:09:30.490 --> 00:09:35.860
to equal the density at that
point, which would be xyz,

00:09:35.860 --> 00:09:39.810
times the volume of that
of that small mass.

00:09:39.810 --> 00:09:42.780
And that volume of that small
mass we could write as dv.

00:09:42.780 --> 00:09:48.790
And we know that dv is the
same thing as the width times

00:09:48.790 --> 00:09:49.670
the height times the depth.

00:09:49.670 --> 00:09:52.350
dv doesn't always have to
be dx times dy times dz.

00:09:52.350 --> 00:09:54.000
If we're doing other
coordinates, if we're doing

00:09:54.000 --> 00:09:57.670
polar coordinates, it could be
something slightly different.

00:09:57.670 --> 00:09:59.160
And we'll do that eventually.

00:09:59.160 --> 00:10:01.280
But if we wanted to figure out
the mass, since we're using

00:10:01.280 --> 00:10:03.550
rectangular coordinates, it
would be the density function

00:10:03.550 --> 00:10:07.030
at that point times our
differential volume.

00:10:07.030 --> 00:10:11.330
So times dx dy dz.

00:10:11.330 --> 00:10:13.870
And of course, we can
change the order here.

00:10:13.870 --> 00:10:16.386
So when you want to figure out
the volume-- when you want to

00:10:16.386 --> 00:10:19.000
figure out the mass-- which I
will do in the next video, we

00:10:19.000 --> 00:10:21.290
essentially will have to
integrate this function.

00:10:21.290 --> 00:10:27.400
As opposed to just
1 over z, y and x.

00:10:27.400 --> 00:10:28.690
And I'm going to do that
in the next video.

00:10:28.690 --> 00:10:32.050
And you'll see that it's really
just a lot of basic taking

00:10:32.050 --> 00:10:34.700
antiderivatives and avoiding
careless mistakes.

00:10:34.700 --> 00:10:37.280
I will see you in
the next video.


Double integrals 4

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:04.430
I think it's very important to
have as many ways as possible

00:00:04.430 --> 00:00:06.450
to view a certain type of
problem, so I want to introduce

00:00:06.450 --> 00:00:07.410
you to a different way.

00:00:07.410 --> 00:00:10.710
Some people might have taught
this first, but the way I

00:00:10.710 --> 00:00:15.850
taught it in the first integral
video is kind of the way that I

00:00:15.850 --> 00:00:17.100
always think about when
I do the problems.

00:00:17.100 --> 00:00:19.330
But sometimes, it's more useful
to think about it the way I'm

00:00:19.330 --> 00:00:21.030
about to show you, and maybe
you won't see the difference,

00:00:21.030 --> 00:00:22.530
or maybe you'll say, oh, Sal,
those are just the

00:00:22.530 --> 00:00:24.010
exact same thing.

00:00:24.010 --> 00:00:26.420
Someone actually emailed me and
told me that I should make it

00:00:26.420 --> 00:00:28.360
so I can scroll things, and I
said, oh, that's not

00:00:28.360 --> 00:00:28.810
too hard to do.

00:00:28.810 --> 00:00:32.140
So I just did that, and
I scrolled my drawing.

00:00:32.140 --> 00:00:34.820
But anyway, let's say we have
a surface in 3 dimensions.

00:00:34.820 --> 00:00:36.120
It's a function of x and y.

00:00:36.120 --> 00:00:38.060
You give me a coordinate down
here, and I'll tell you

00:00:38.060 --> 00:00:40.240
how high the surface
is at that point.

00:00:40.240 --> 00:00:43.330
And we want to figure out the
volume under that surface.

00:00:43.330 --> 00:00:44.880
So.

00:00:44.880 --> 00:00:49.360
We can very easily figure out
the volume of a very small

00:00:49.360 --> 00:00:52.210
column underneath the surface.

00:00:52.210 --> 00:00:55.870
So this whole volume is what
we're trying to figure out,

00:00:55.870 --> 00:00:57.020
right, between the
dotted lines.

00:00:57.020 --> 00:00:59.050
I think you can see it.

00:00:59.050 --> 00:01:01.730
You have some experience
visualizing this right now.

00:01:01.730 --> 00:01:05.940
So let's say that I have
a little area here.

00:01:05.940 --> 00:01:09.550
We could call that da.

00:01:09.550 --> 00:01:12.240
Let me see if I can draw this.

00:01:12.240 --> 00:01:15.430
Let's say we have a little
area down here, a little

00:01:15.430 --> 00:01:17.780
square in the x-y plane.

00:01:17.780 --> 00:01:23.580
And it's, depending on how you
view it, this side of it is dx,

00:01:23.580 --> 00:01:27.480
its length is dx, and the
height, you could say,

00:01:27.480 --> 00:01:29.700
on that side, is dy.

00:01:29.700 --> 00:01:30.390
Right?

00:01:30.390 --> 00:01:33.100
Because it's a little small
change in y there, and it's a

00:01:33.100 --> 00:01:36.490
little small change in x here.

00:01:36.490 --> 00:01:40.060
And its area, the area of
this little square, is

00:01:40.060 --> 00:01:42.800
going to be dx times dy.

00:01:42.800 --> 00:01:47.720
And if we wanted to figure out
the volume of the solid between

00:01:47.720 --> 00:01:52.090
this little area and the
surface, we could just multiply

00:01:52.090 --> 00:01:54.940
this area times the function.

00:01:54.940 --> 00:01:55.250
Right?

00:01:55.250 --> 00:01:57.210
Because the height at this
point is going to be the

00:01:57.210 --> 00:01:59.090
value of the function,
roughly, at this point.

00:01:59.090 --> 00:01:59.300
Right?

00:01:59.300 --> 00:02:00.640
This is going to be an
approximation, and then we're

00:02:00.640 --> 00:02:01.450
going to take an infinite sum.

00:02:01.450 --> 00:02:03.220
I think you know
where this is going.

00:02:03.220 --> 00:02:04.170
But let me do that.

00:02:04.170 --> 00:02:10.480
Let me at least draw the little
column that I want to show you.

00:02:10.480 --> 00:02:16.690
So that's one end of it, that's
another end of it, that's the

00:02:16.690 --> 00:02:21.840
front end of it, that's
the other end of it.

00:02:21.840 --> 00:02:25.660
So we have a little figure that
looks something like that.

00:02:25.660 --> 00:02:28.180
A little column, right?

00:02:28.180 --> 00:02:30.400
It intersects the
top of the surface.

00:02:30.400 --> 00:02:33.950
And the volume of this
column, not too difficult.

00:02:33.950 --> 00:02:38.420
It's going to be this little
area down here, which is,

00:02:38.420 --> 00:02:39.360
we could call that da.

00:02:41.870 --> 00:02:43.300
Sometimes written
like that. da.

00:02:43.300 --> 00:02:44.565
It's a little area.

00:02:44.565 --> 00:02:47.890
And we're going to multiply
that area times the height of

00:02:47.890 --> 00:02:51.280
this column, and that's the
function at that point.

00:02:51.280 --> 00:02:56.260
So it's f of x and y.

00:02:56.260 --> 00:02:58.970
And of course, we could have
also written it as, this

00:02:58.970 --> 00:03:02.590
da is just dx times
dy, or dy times dx.

00:03:02.590 --> 00:03:04.920
I'm going to write it in
every different way.

00:03:04.920 --> 00:03:08.260
So we could also have
written this as f of

00:03:08.260 --> 00:03:12.510
xy times dx times dy.

00:03:12.510 --> 00:03:14.730
And of course, since
multiplication is associative,

00:03:14.730 --> 00:03:21.140
I could have also written it as
f of xy times dy dx.

00:03:21.140 --> 00:03:24.270
These are all equivalent, and
these all represent the volume

00:03:24.270 --> 00:03:29.010
of this column, that's the
between this little area

00:03:29.010 --> 00:03:31.550
here and the surface.

00:03:31.550 --> 00:03:35.980
So now, if we wanted to figure
out the volume of the entire

00:03:35.980 --> 00:03:37.850
surface, we have a couple
of things we could do.

00:03:37.850 --> 00:03:42.120
We could add up all of the
volumes in the x-direction,

00:03:42.120 --> 00:03:45.230
between the lower x-bound and
the upper x-bound, and then

00:03:45.230 --> 00:03:49.330
we'd have kind of a thin sheet,
although it will already have

00:03:49.330 --> 00:03:51.710
some depth, because we're
not adding up just the x's.

00:03:51.710 --> 00:03:53.060
There's also a dy back there.

00:03:53.060 --> 00:03:56.870
So we would have a volume of a
figure that would extend from

00:03:56.870 --> 00:04:03.170
the lower x all the way to the
upper x, go back dy,

00:04:03.170 --> 00:04:04.210
and come back here.

00:04:04.210 --> 00:04:07.390
If we wanted to sum
up all the dx's.

00:04:07.390 --> 00:04:11.050
And if we wanted to do that,
which expression would we use?

00:04:11.050 --> 00:04:15.800
Well, we would be summing with
respect to x first, so we could

00:04:15.800 --> 00:04:17.040
use this expression, right?

00:04:19.630 --> 00:04:21.035
And actually, we could
write it here, but it

00:04:21.035 --> 00:04:21.970
just becomes confusing.

00:04:21.970 --> 00:04:24.880
If we're summing with respect
to x, but we have the

00:04:24.880 --> 00:04:26.110
dy written here first.

00:04:26.110 --> 00:04:28.430
It's really not incorrect, but
it just becomes a little

00:04:28.430 --> 00:04:30.740
ambiguous, are we summing
with respect to x or y.

00:04:30.740 --> 00:04:32.430
But here, we could say, OK.

00:04:32.430 --> 00:04:36.140
If we want to sum up all the
dx's first, let's do that.

00:04:36.140 --> 00:04:38.890
We're taking the sum with
respect to x, and let me, I'm

00:04:38.890 --> 00:04:41.270
going to write down the actual,
normally I just write numbers

00:04:41.270 --> 00:04:44.210
here, but I'm going to say,
well, the lower bound here is x

00:04:44.210 --> 00:04:51.320
is equal to a, and the upper
bound here is x is equal to b.

00:04:54.650 --> 00:04:57.380
And that'll give us the volume
of, you could imagine a

00:04:57.380 --> 00:04:59.330
sheet with depth, right?

00:04:59.330 --> 00:05:03.760
The sheet is going to be
parallel to the x-axis, right?

00:05:03.760 --> 00:05:07.060
And then once we have that
sheet, in my video, I think

00:05:07.060 --> 00:05:10.280
that's the newspaper people
trying to sell me something.

00:05:10.280 --> 00:05:11.040
Anyway.

00:05:11.040 --> 00:05:13.470
So once we have the sheet, I'll
try to draw it here, too, I

00:05:13.470 --> 00:05:16.930
don't want to get this picture
too muddied up, but once we

00:05:16.930 --> 00:05:20.010
have that sheet, then we can
integrate those, we can

00:05:20.010 --> 00:05:21.360
add up the dy's, right?

00:05:21.360 --> 00:05:23.480
Because this width right
here is still dy.

00:05:23.480 --> 00:05:25.300
We could add up of all the
different dy's, and we

00:05:25.300 --> 00:05:27.390
would have the volume
of the whole figure.

00:05:27.390 --> 00:05:30.910
So once we take this sum,
then we could take this sum.

00:05:30.910 --> 00:05:35.320
Where y is going from it's
bottom, which we said with c,

00:05:35.320 --> 00:05:39.230
from y is equal to c to y's
upper bound, to y

00:05:39.230 --> 00:05:42.070
is equal to d.

00:05:42.070 --> 00:05:42.690
Fair enough.

00:05:42.690 --> 00:05:45.340
And then, once we evaluate
this whole thing, we have the

00:05:45.340 --> 00:05:49.520
volume of this solid, or the
volume under the surface.

00:05:49.520 --> 00:05:51.330
Now we could have
gone the other way.

00:05:51.330 --> 00:05:53.810
I know this gets a little
bit messy, but I think

00:05:53.810 --> 00:05:54.840
you get what I'm saying.

00:05:54.840 --> 00:05:57.185
Let's start with that little
da we had originally.

00:06:00.280 --> 00:06:04.520
Instead of going this way,
instead of summing up the dx's

00:06:04.520 --> 00:06:08.910
and getting this sheet, let's
sum up the dy's first, right?

00:06:08.910 --> 00:06:12.570
So we could take, we're summing
in the y-direction first.

00:06:12.570 --> 00:06:16.420
We would get a sheet that's
parallel to the y-axis, now.

00:06:16.420 --> 00:06:21.770
So the top of the sheet would
look something like that.

00:06:21.770 --> 00:06:25.705
So if we're coming the dy's
first, we would take the sum,

00:06:25.705 --> 00:06:28.780
we would take the integral with
respect to y, and it would be,

00:06:28.780 --> 00:06:33.460
the lower bound would be y is
equal to c, and the upper

00:06:33.460 --> 00:06:36.600
bound is y is equal to d.

00:06:36.600 --> 00:06:39.346
And then we would have that
sheet with a little depth, the

00:06:39.346 --> 00:06:44.790
depth is dx, and then we could
take the sum of all of those,

00:06:44.790 --> 00:06:46.580
sorry, my throat is dry.

00:06:46.580 --> 00:06:50.290
I just had a bunch of almonds
to get power to be able

00:06:50.290 --> 00:06:52.070
to record these videos.

00:06:52.070 --> 00:06:54.230
But once I have one of these
sheets, and if I want to sum up

00:06:54.230 --> 00:06:58.180
all of the x's, then I could
take the infinite sum of

00:06:58.180 --> 00:07:02.520
infinitely small columns, or in
this view, sheets, infinitely

00:07:02.520 --> 00:07:08.200
small depths, and the lower
bound is x is equal to a, and

00:07:08.200 --> 00:07:10.100
the upper bound is
x is equal to b.

00:07:10.100 --> 00:07:12.660
And once again, I would have
the volume of the figure.

00:07:12.660 --> 00:07:15.930
And all I showed you here is
that there's two ways of doing

00:07:15.930 --> 00:07:17.790
the order of integration.

00:07:17.790 --> 00:07:21.060
Now, another way of saying
this, if this little original

00:07:21.060 --> 00:07:24.190
square was da, and this is a
shorthand that you'll see all

00:07:24.190 --> 00:07:27.300
the time, especially in
physics textbooks, is that we

00:07:27.300 --> 00:07:33.440
are integrating along
the domain, right?

00:07:33.440 --> 00:07:35.770
Because the x-y plane
here is our domain.

00:07:35.770 --> 00:07:37.690
So we're going to do a double
integral, a two-dimensional

00:07:37.690 --> 00:07:40.132
integral, we're saying that the
domain here is two-dimensional,

00:07:40.132 --> 00:07:48.110
and we're going to take that
over f of x and y times da.

00:07:48.110 --> 00:07:51.240
And the reason why I want to
show you this, is you see this

00:07:51.240 --> 00:07:53.830
in physics books all the time.

00:07:53.830 --> 00:07:55.300
I don't think it's a
great thing to do.

00:07:55.300 --> 00:07:58.860
Because it is a shorthand, and
maybe it looks simpler, but for

00:07:58.860 --> 00:08:00.920
me, whenever I see something
that I don't know how to

00:08:00.920 --> 00:08:04.180
compute or that's not obvious
for me to know how to compute,

00:08:04.180 --> 00:08:06.320
it actually is more confusing.

00:08:06.320 --> 00:08:09.260
So I wanted to just show you
that what you see in this

00:08:09.260 --> 00:08:12.080
physics book, when someone
writes this, it's the exact

00:08:12.080 --> 00:08:14.350
same thing as this or this.

00:08:14.350 --> 00:08:18.170
The da could either be dx times
dy, or it could either be dy

00:08:18.170 --> 00:08:21.420
times dx, and when they do this
double integral over domain,

00:08:21.420 --> 00:08:24.380
that's the same thing is just
adding up all of these squares.

00:08:24.380 --> 00:08:26.550
Where we do it here, we're
very ordered about it, right?

00:08:26.550 --> 00:08:30.580
We go in the x-direction, and
then we add all of those up in

00:08:30.580 --> 00:08:32.350
the y-direction, and we
get the entire volume.

00:08:32.350 --> 00:08:34.020
Or we could go the
other way around.

00:08:34.020 --> 00:08:36.580
When we say that we're just
taking the double integral,

00:08:36.580 --> 00:08:37.950
first of all, that tells us
we're doing it in two

00:08:37.950 --> 00:08:40.710
dimensions, over a domain,
that leaves it a little bit

00:08:40.710 --> 00:08:43.790
ambiguous in terms of
how we're going to sum

00:08:43.790 --> 00:08:45.430
up all of the da's.

00:08:45.430 --> 00:08:47.990
And they do it intentionally in
physics books, because you

00:08:47.990 --> 00:08:50.750
don't have to do it using
Cartesian coordinates,

00:08:50.750 --> 00:08:51.720
using x's and y's.

00:08:51.720 --> 00:08:53.940
You can do it in polar
coordinates, you could do it

00:08:53.940 --> 00:08:55.260
a ton of different ways.

00:08:55.260 --> 00:08:58.200
But I just wanted to show
you, this is another way to

00:08:58.200 --> 00:09:00.870
having an intuition of the
volume under a surface.

00:09:00.870 --> 00:09:04.620
And these are the exact same
things as this type of

00:09:04.620 --> 00:09:07.840
notation that you might
see in a physics book.

00:09:07.840 --> 00:09:09.420
Sometimes they won't write
a domain, sometimes they'd

00:09:09.420 --> 00:09:11.250
write over a surface.

00:09:11.250 --> 00:09:12.430
And we'll later do
those integrals.

00:09:12.430 --> 00:09:15.080
Here the surface is easy, it's
a flat plane, but sometimes

00:09:15.080 --> 00:09:17.380
it'll end up being a curve
or something like that.

00:09:17.380 --> 00:09:19.790
But anyway, I'm
almost out of time.

00:09:19.790 --> 00:09:22.390
I will see you in
the next video.


Double integrals 6

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.430
Welcome back.

00:00:01.430 --> 00:00:04.870
In the last video we were just
figuring out the volume under

00:00:04.870 --> 00:00:06.990
the surface, and we had set
up these integral bounds.

00:00:06.990 --> 00:00:09.280
So let's see how to
evaluate it now.

00:00:09.280 --> 00:00:10.400
And look at this.

00:00:10.400 --> 00:00:13.510
I actually realized that I can
scroll things, which is quite

00:00:13.510 --> 00:00:16.790
useful because now I have
a lot more board space.

00:00:16.790 --> 00:00:18.290
So how do we evaluate
this integral?

00:00:18.290 --> 00:00:22.030
Well, the first integral I'm
integrating with respect to x.

00:00:22.030 --> 00:00:22.280
Right?

00:00:22.280 --> 00:00:24.770
I'm adding up the
little x sums.

00:00:24.770 --> 00:00:27.670
So I'm forming this
rectangle right here.

00:00:27.670 --> 00:00:27.960
Right?

00:00:27.960 --> 00:00:30.550
Or you could kind of view it
I'm holding y constant and

00:00:30.550 --> 00:00:32.730
integrating along the x-axis.

00:00:32.730 --> 00:00:34.630
I should switch colors.

00:00:34.630 --> 00:00:38.660
So what's the antiderivative of
x y squared with respect to x?

00:00:38.660 --> 00:00:44.000
Well it's just x
squared over 2.

00:00:44.000 --> 00:00:46.080
And then I have the y
squared-- that's just a

00:00:46.080 --> 00:00:48.900
constant-- all over 2.

00:00:48.900 --> 00:00:55.890
And I'm going to evaluate that
from x is equal to 1 to x is

00:00:55.890 --> 00:00:58.660
equal to the square root of y,
which you might be daunted by.

00:00:58.660 --> 00:01:00.930
But you'll see that it's
actually not that bad

00:01:00.930 --> 00:01:01.630
once you evaluate them.

00:01:01.630 --> 00:01:03.540
And then let me draw the
outside of the integral.

00:01:03.540 --> 00:01:06.620
This is y is equal to
0 to y is equal 1.

00:01:06.620 --> 00:01:08.830
dy.

00:01:08.830 --> 00:01:11.640
Now, if x is equal to 1
this expression becomes

00:01:11.640 --> 00:01:13.400
y squared over 2.

00:01:13.400 --> 00:01:14.470
Right?

00:01:14.470 --> 00:01:19.030
y squared over 2, minus-- now
if x is equal to square root

00:01:19.030 --> 00:01:21.320
of y, what does this
expression become?

00:01:21.320 --> 00:01:25.120
If x is equal to the
square root of y, then

00:01:25.120 --> 00:01:27.250
x squared is just y.

00:01:27.250 --> 00:01:30.870
And then y times y squared
is y to the third.

00:01:30.870 --> 00:01:31.180
Right?

00:01:31.180 --> 00:01:36.210
So it's y to the third over 3.

00:01:36.210 --> 00:01:36.830
Fair enough.

00:01:36.830 --> 00:01:40.870
And now I take the integral
with respect to y.

00:01:40.870 --> 00:01:45.520
So now I sum up all of these
rectangles in the y direction.

00:01:45.520 --> 00:01:46.420
0, 1.

00:01:46.420 --> 00:01:47.620
This is with respect to y.

00:01:47.620 --> 00:01:48.910
And that's cool, right?

00:01:48.910 --> 00:01:52.820
Because when you take the first
integral with respect to x you

00:01:52.820 --> 00:01:55.030
end up with a function of y
anyway, so you might as well

00:01:55.030 --> 00:01:57.120
have your bounds as
functions of y's.

00:01:57.120 --> 00:01:59.510
It really doesn't make
it any more difficult.

00:01:59.510 --> 00:02:00.850
But anyway, back
to the problem.

00:02:00.850 --> 00:02:07.440
What is the antiderivative
of y squared over 2 minus

00:02:07.440 --> 00:02:11.380
y to the third over 3?

00:02:11.380 --> 00:02:15.430
Well the antiderivative of y
squared-- and you have to

00:02:15.430 --> 00:02:18.130
divide by 3, so it's
y cubed over 6.

00:02:18.130 --> 00:02:23.670
Minus y to the fourth--
you have to divide by 4.

00:02:23.670 --> 00:02:28.480
Minus y to the fourth over--
did I mess up some place?

00:02:28.480 --> 00:02:32.385
No, I think this is correct.
y to the fourth over 12.

00:02:36.080 --> 00:02:36.530
Oh wait.

00:02:36.530 --> 00:02:37.480
How did I get a 3 here?

00:02:37.480 --> 00:02:38.330
That's where I messed up.

00:02:38.330 --> 00:02:39.360
This is a 2, right?

00:02:44.490 --> 00:02:45.520
Let's see.

00:02:45.520 --> 00:02:46.800
x is square root of y.

00:02:46.800 --> 00:02:48.270
Yeah, this is a 2.

00:02:48.270 --> 00:02:49.150
I don't know how I ended up.

00:02:49.150 --> 00:02:49.510
Right?

00:02:49.510 --> 00:02:52.270
Square root of y squared
is y, times y squared

00:02:52.270 --> 00:02:53.575
y to the third over 2.

00:02:57.470 --> 00:02:58.030
Right.

00:02:58.030 --> 00:03:00.900
And then when I take
the integral of this

00:03:00.900 --> 00:03:03.590
it's 4 times 2.

00:03:03.590 --> 00:03:04.240
8.

00:03:04.240 --> 00:03:06.670
Got to make sure I don't make
those careless mistakes.

00:03:06.670 --> 00:03:07.440
That's the tough part.

00:03:07.440 --> 00:03:09.500
I just want to make sure
that you got that too.

00:03:09.500 --> 00:03:10.620
I hate it when I do that.

00:03:10.620 --> 00:03:12.350
But I don't want to
re-record the whole video.

00:03:12.350 --> 00:03:15.340
So when I evaluated
this-- right.

00:03:15.340 --> 00:03:17.490
This is right, and then I take
the antiderivative of y to the

00:03:17.490 --> 00:03:21.430
third over 2, I get y
to the fourth over 8.

00:03:21.430 --> 00:03:26.550
And now I evaluate
this at 1 and 0.

00:03:26.550 --> 00:03:27.540
And that give us what?

00:03:27.540 --> 00:03:34.290
1/6 minus 1/8 minus-- well both
of these when you evaluate

00:03:34.290 --> 00:03:35.410
them, are going to be 0's.

00:03:35.410 --> 00:03:37.700
So this is going to be
another 0 minus 0.

00:03:37.700 --> 00:03:38.530
So you don't have to
worry about that.

00:03:38.530 --> 00:03:41.000
So what's 1/6 minus 1/8?

00:03:41.000 --> 00:03:41.930
Let's see.

00:03:41.930 --> 00:03:43.310
24.

00:03:43.310 --> 00:03:50.070
That's 4 minus 3 over 24, which
is equal to 1/24-- is the

00:03:50.070 --> 00:03:51.600
volume of our figure.

00:03:55.640 --> 00:04:00.280
So this time, the way we just
did it, we took the integral

00:04:00.280 --> 00:04:02.486
with respect to x first, and
then we did it with

00:04:02.486 --> 00:04:02.950
respect to y.

00:04:02.950 --> 00:04:04.930
Let's do it the
other way around.

00:04:04.930 --> 00:04:07.520
So let me erase some things.

00:04:07.520 --> 00:04:11.390
And hopefully I won't make
these careless mistakes again.

00:04:11.390 --> 00:04:14.670
I'll keep this figure, but
I'll even erase this one.

00:04:19.486 --> 00:04:21.704
Let me erase all of this stuff.

00:04:24.280 --> 00:04:27.430
We have room to work with.

00:04:27.430 --> 00:04:29.150
So I kept that figure.

00:04:29.150 --> 00:04:33.510
But let me redraw just the
xy plane, just so we get

00:04:33.510 --> 00:04:36.300
the visualization right.

00:04:36.300 --> 00:04:38.400
It's more important to
visualize the xy plane in

00:04:38.400 --> 00:04:41.070
these problems than it
is to visualize the

00:04:41.070 --> 00:04:44.090
whole thing in 3-d.

00:04:44.090 --> 00:04:46.350
That's the y-axis,
that's the x-axis.

00:04:51.870 --> 00:04:52.230
OK.

00:04:52.230 --> 00:04:55.720
Our upper bound, you can
say, is the graph y is

00:04:55.720 --> 00:04:56.460
equal to x squared.

00:04:59.210 --> 00:05:00.848
Or, you could view it as
the bound x is equal to

00:05:00.848 --> 00:05:02.000
the square root of y.

00:05:02.000 --> 00:05:06.510
That is x is equal to 1,
that's y is equal to 1.

00:05:06.510 --> 00:05:09.930
And we care about the volume
above the shaded region.

00:05:09.930 --> 00:05:10.190
Right?

00:05:10.190 --> 00:05:12.750
That shaded region is this
yellow region right here.

00:05:12.750 --> 00:05:15.800
And let's draw our da.

00:05:15.800 --> 00:05:18.520
I'll draw a little
square actually.

00:05:18.520 --> 00:05:20.025
And I'll do it in magenta.

00:05:23.490 --> 00:05:26.190
So that's our little da.

00:05:26.190 --> 00:05:29.160
And the height is dy.

00:05:29.160 --> 00:05:30.370
That's a y.

00:05:30.370 --> 00:05:31.260
dy.

00:05:31.260 --> 00:05:33.240
And it's with this dx.

00:05:33.240 --> 00:05:33.870
Right?

00:05:33.870 --> 00:05:38.290
So the volume above this little
square-- that's the same

00:05:38.290 --> 00:05:39.680
thing as this little square.

00:05:39.680 --> 00:05:42.620
Just like we said before, the
volume above it is equal to

00:05:42.620 --> 00:05:43.850
the value of the function.

00:05:43.850 --> 00:05:44.090
Right?

00:05:44.090 --> 00:05:49.210
The height is the value of the
function, which is x y squared.

00:05:49.210 --> 00:05:51.390
And then we multiply it
times the area of the base.

00:05:51.390 --> 00:05:54.700
Well the area of the base, you
could say it's da, but we know

00:05:54.700 --> 00:05:56.490
it's really dy times dx.

00:05:59.100 --> 00:06:00.680
I didn't have to write
that times there.

00:06:00.680 --> 00:06:02.450
You can ignore it.

00:06:02.450 --> 00:06:03.800
And I wrote the y first
just because we're

00:06:03.800 --> 00:06:05.490
going to integrate with
respect to y first.

00:06:05.490 --> 00:06:07.310
We're going to sum
in the y direction.

00:06:07.310 --> 00:06:09.300
So what does summing in
the y direction mean?

00:06:09.300 --> 00:06:11.920
It means we're going to
add that square to that

00:06:11.920 --> 00:06:14.828
square to that square--

00:06:14.828 --> 00:06:16.070
[COUGHING]

00:06:16.070 --> 00:06:16.800
Excuse me.

00:06:16.800 --> 00:06:21.290
So we're adding all the
dy's together, right?

00:06:21.290 --> 00:06:26.230
So my question to you is, what
is the upper bound on the y?

00:06:26.230 --> 00:06:28.600
Well once again, we
bump into the curve.

00:06:28.600 --> 00:06:28.970
Right?

00:06:28.970 --> 00:06:32.000
So the curve is the upper
bound when we go upwards.

00:06:32.000 --> 00:06:34.130
And what is the upper
bound on the curve?

00:06:34.130 --> 00:06:37.680
Well we're holding x
fixed, so for any given

00:06:37.680 --> 00:06:40.300
x what is this point?

00:06:40.300 --> 00:06:41.730
Well that's going
to be x squared.

00:06:41.730 --> 00:06:42.000
Right?

00:06:42.000 --> 00:06:43.940
Because this is the graph
of y equals x squared.

00:06:43.940 --> 00:06:48.700
So our upper bound is y
is equal to x squared.

00:06:48.700 --> 00:06:49.800
And what's our lower bound?

00:06:49.800 --> 00:06:52.310
We can keep adding these
squares down here.

00:06:52.310 --> 00:06:54.550
We're adding all the
little changes in y.

00:06:54.550 --> 00:06:55.610
So what's our lower bound?

00:06:55.610 --> 00:06:56.730
Well our lower bound is just 0.

00:06:56.730 --> 00:06:57.575
That was pretty
straightforward.

00:07:00.470 --> 00:07:06.600
So this expression, as it is
written right now, is the

00:07:06.600 --> 00:07:10.110
volume above this rectangle.

00:07:10.110 --> 00:07:11.940
Right?

00:07:11.940 --> 00:07:13.790
Let me draw it.

00:07:13.790 --> 00:07:15.860
Is the volume above
this rectangle.

00:07:15.860 --> 00:07:18.890
These are the same rectangles.

00:07:18.890 --> 00:07:22.730
The volume above
that rectangle.

00:07:22.730 --> 00:07:25.790
Now what we want to do is add
up all the dx's together, and

00:07:25.790 --> 00:07:27.670
we'll get the volume above
the entire surface.

00:07:27.670 --> 00:07:29.880
So that rectangle, now
we'll add it to another

00:07:29.880 --> 00:07:32.240
dx one, dx one like that.

00:07:32.240 --> 00:07:34.920
So what's the upper and
lower bounds on the x's?

00:07:34.920 --> 00:07:37.070
We're going from x is
equal to 0, right?

00:07:37.070 --> 00:07:39.650
We don't bump into the graph
when we go all the way down.

00:07:39.650 --> 00:07:41.950
So we go from x is equal to 0.

00:07:41.950 --> 00:07:45.060
And then our upper bound
is x is equal to 1.

00:07:45.060 --> 00:07:47.520
So x is equal to 0,
x is equal to 1.

00:07:47.520 --> 00:07:51.630
And in general, one way to
think about it is when you're

00:07:51.630 --> 00:07:56.020
doing kind of the last sum or
the last integral, you really

00:07:56.020 --> 00:07:59.160
shouldn't have variable
boundaries at this point.

00:07:59.160 --> 00:07:59.520
Right?

00:07:59.520 --> 00:08:03.120
Because our final answer has
to be a number, assuming

00:08:03.120 --> 00:08:06.360
that we're not dealing with
something very, very abstract.

00:08:06.360 --> 00:08:07.980
But our final answer is
going to have a number.

00:08:07.980 --> 00:08:09.760
So if you had some variables
here you probably

00:08:09.760 --> 00:08:10.740
did something wrong.

00:08:10.740 --> 00:08:13.640
And it's always useful, I
think, to draw that little da

00:08:13.640 --> 00:08:16.160
and then figure out-- OK,
I'm summing to dy first.

00:08:16.160 --> 00:08:18.320
When I go upwards I
bump into the curve.

00:08:18.320 --> 00:08:20.210
What is that upper bound
if x is constant?

00:08:20.210 --> 00:08:22.380
Oh, it's x squared. y
is equal to x squared.

00:08:22.380 --> 00:08:26.260
If I go down I bump into the
x-axis, or y is equal to 0.

00:08:26.260 --> 00:08:27.410
And so forth and so on.

00:08:27.410 --> 00:08:29.910
So now let's just evaluate
this and confirm that

00:08:29.910 --> 00:08:32.110
we get the same answer.

00:08:32.110 --> 00:08:34.600
So we're integrating with
respect to y first.

00:08:34.600 --> 00:08:40.040
So that's x y to the
third, over three.

00:08:40.040 --> 00:08:42.990
At x squared and 0.

00:08:42.990 --> 00:08:46.050
And then we have our outer
integral. x goes from 0 to 1.

00:08:46.050 --> 00:08:48.050
dx.

00:08:48.050 --> 00:08:51.490
If we substitute x
squared in for y.

00:08:51.490 --> 00:08:55.190
x squared to the third
power is x to the sixth.

00:08:55.190 --> 00:08:58.800
x to the sixth times
x-- let me write that.

00:08:58.800 --> 00:09:04.300
So we have x times x squared
to the third power, over 3.

00:09:04.300 --> 00:09:08.540
That equals x to the seventh.

00:09:08.540 --> 00:09:11.750
Right? x squared to the third--
you multiply the exponents, and

00:09:11.750 --> 00:09:16.350
then you add these. x to the
seventh over 3, minus this

00:09:16.350 --> 00:09:17.790
evaluated with y is 0.

00:09:17.790 --> 00:09:18.940
But that's just going to be 0.

00:09:18.940 --> 00:09:20.410
Right?

00:09:20.410 --> 00:09:26.400
And then we evaluate
that from 0 to 1, dx.

00:09:26.400 --> 00:09:28.680
We're almost there.

00:09:28.680 --> 00:09:30.620
Increment the exponent.

00:09:30.620 --> 00:09:33.370
You get x to the eighth over 8.

00:09:33.370 --> 00:09:36.950
And we already have a 3 down
there, so it's over 24.

00:09:36.950 --> 00:09:39.760
And you evaluate
that from 0 to 1.

00:09:39.760 --> 00:09:42.290
And I think we get
the same answer.

00:09:42.290 --> 00:09:46.100
When you evaluate it at
1 you get 1/24 minus 0.

00:09:46.100 --> 00:09:49.180
So once again, when we
integrate in the other order

00:09:49.180 --> 00:09:53.340
you still get the volume of
the figure, being 1/4,

00:09:53.340 --> 00:09:55.290
whatever, cubic units.

00:09:55.290 --> 00:09:57.760
Anyway, see you in
the next video.