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# Double integrals 4

Another way to conceptualize the double integral. Created by Sal Khan.

Video transcript

I think it's very important to
have as many ways as possible to view a certain type of
problem, so I want to introduce you to a different way. Some people might have taught
this first, but the way I taught it in the first integral
video is kind of the way that I always think about when
I do the problems. But sometimes, it's more useful
to think about it the way I'm about to show you, and maybe
you won't see the difference, or maybe you'll say, oh, Sal,
those are just the exact same thing. Someone actually emailed me and
told me that I should make it so I can scroll things, and I
said, oh, that's not too hard to do. So I just did that, and
I scrolled my drawing. But anyway, let's say we have
a surface in 3 dimensions. It's a function of x and y. You give me a coordinate down
here, and I'll tell you how high the surface
is at that point. And we want to figure out the
volume under that surface. So. We can very easily figure out
the volume of a very small column underneath the surface. So this whole volume is what
we're trying to figure out, right, between the
dotted lines. I think you can see it. You have some experience
visualizing this right now. So let's say that I have
a little area here. We could call that da. Let me see if I can draw this. Let's say we have a little
area down here, a little square in the x-y plane. And it's, depending on how you
view it, this side of it is dx, its length is dx, and the
height, you could say, on that side, is dy. Right? Because it's a little small
change in y there, and it's a little small change in x here. And its area, the area of
this little square, is going to be dx times dy. And if we wanted to figure out
the volume of the solid between this little area and the
surface, we could just multiply this area times the function. Right? Because the height at this
point is going to be the value of the function,
roughly, at this point. Right? This is going to be an
approximation, and then we're going to take an infinite sum. I think you know
where this is going. But let me do that. Let me at least draw the little
column that I want to show you. So that's one end of it, that's
another end of it, that's the front end of it, that's
the other end of it. So we have a little figure that
looks something like that. A little column, right? It intersects the
top of the surface. And the volume of this
column, not too difficult. It's going to be this little
area down here, which is, we could call that da. Sometimes written
like that. da. It's a little area. And we're going to multiply
that area times the height of this column, and that's the
function at that point. So it's f of x and y. And of course, we could have
also written it as, this da is just dx times
dy, or dy times dx. I'm going to write it in
every different way. So we could also have
written this as f of xy times dx times dy. And of course, since
multiplication is associative, I could have also written it as
f of xy times dy dx. These are all equivalent, and
these all represent the volume of this column, that's the
between this little area here and the surface. So now, if we wanted to figure
out the volume of the entire surface, we have a couple
of things we could do. We could add up all of the
volumes in the x-direction, between the lower x-bound and
the upper x-bound, and then we'd have kind of a thin sheet,
although it will already have some depth, because we're
not adding up just the x's. There's also a dy back there. So we would have a volume of a
figure that would extend from the lower x all the way to the
upper x, go back dy, and come back here. If we wanted to sum
up all the dx's. And if we wanted to do that,
which expression would we use? Well, we would be summing with
respect to x first, so we could use this expression, right? And actually, we could
write it here, but it just becomes confusing. If we're summing with respect
to x, but we have the dy written here first. It's really not incorrect, but
it just becomes a little ambiguous, are we summing
with respect to x or y. But here, we could say, OK. If we want to sum up all the
dx's first, let's do that. We're taking the sum with
respect to x, and let me, I'm going to write down the actual,
normally I just write numbers here, but I'm going to say,
well, the lower bound here is x is equal to a, and the upper
bound here is x is equal to b. And that'll give us the volume
of, you could imagine a sheet with depth, right? The sheet is going to be
parallel to the x-axis, right? And then once we have that
sheet, in my video, I think that's the newspaper people
trying to sell me something. Anyway. So once we have the sheet, I'll
try to draw it here, too, I don't want to get this picture
too muddied up, but once we have that sheet, then we can
integrate those, we can add up the dy's, right? Because this width right
here is still dy. We could add up of all the
different dy's, and we would have the volume
of the whole figure. So once we take this sum,
then we could take this sum. Where y is going from it's
bottom, which we said with c, from y is equal to c to y's
upper bound, to y is equal to d. Fair enough. And then, once we evaluate
this whole thing, we have the volume of this solid, or the
volume under the surface. Now we could have
gone the other way. I know this gets a little
bit messy, but I think you get what I'm saying. Let's start with that little
da we had originally. Instead of going this way,
instead of summing up the dx's and getting this sheet, let's
sum up the dy's first, right? So we could take, we're summing
in the y-direction first. We would get a sheet that's
parallel to the y-axis, now. So the top of the sheet would
look something like that. So if we're coming the dy's
first, we would take the sum, we would take the integral with
respect to y, and it would be, the lower bound would be y is
equal to c, and the upper bound is y is equal to d. And then we would have that
sheet with a little depth, the depth is dx, and then we could
take the sum of all of those, sorry, my throat is dry. I just had a bunch of almonds
to get power to be able to record these videos. But once I have one of these
sheets, and if I want to sum up all of the x's, then I could
take the infinite sum of infinitely small columns, or in
this view, sheets, infinitely small depths, and the lower
bound is x is equal to a, and the upper bound is
x is equal to b. And once again, I would have
the volume of the figure. And all I showed you here is
that there's two ways of doing the order of integration. Now, another way of saying
this, if this little original square was da, and this is a
shorthand that you'll see all the time, especially in
physics textbooks, is that we are integrating along
the domain, right? Because the x-y plane
here is our domain. So we're going to do a double
integral, a two-dimensional integral, we're saying that the
domain here is two-dimensional, and we're going to take that
over f of x and y times da. And the reason why I want to
show you this, is you see this in physics books all the time. I don't think it's a
great thing to do. Because it is a shorthand, and
maybe it looks simpler, but for me, whenever I see something
that I don't know how to compute or that's not obvious
for me to know how to compute, it actually is more confusing. So I wanted to just show you
that what you see in this physics book, when someone
writes this, it's the exact same thing as this or this. The da could either be dx times
dy, or it could either be dy times dx, and when they do this
double integral over domain, that's the same thing is just
adding up all of these squares. Where we do it here, we're
very ordered about it, right? We go in the x-direction, and
then we add all of those up in the y-direction, and we
get the entire volume. Or we could go the
other way around. When we say that we're just
taking the double integral, first of all, that tells us
we're doing it in two dimensions, over a domain,
that leaves it a little bit ambiguous in terms of
how we're going to sum up all of the da's. And they do it intentionally in
physics books, because you don't have to do it using
Cartesian coordinates, using x's and y's. You can do it in polar
coordinates, you could do it a ton of different ways. But I just wanted to show
you, this is another way to having an intuition of the
volume under a surface. And these are the exact same
things as this type of notation that you might
see in a physics book. Sometimes they won't write
a domain, sometimes they'd write over a surface. And we'll later do
those integrals. Here the surface is easy, it's
a flat plane, but sometimes it'll end up being a curve
or something like that. But anyway, I'm
almost out of time. I will see you in
the next video.