Main content

## Multivariable calculus

### Course: Multivariable calculus > Unit 4

Lesson 11: Surface integrals- Introduction to the surface integral
- Find area elements
- Example of calculating a surface integral part 1
- Example of calculating a surface integral part 2
- Example of calculating a surface integral part 3
- Surface integrals to find surface area
- Surface integral example, part 1
- Surface integral example part 2
- Surface integral example part 3: The home stretch
- Surface integral ex2 part 1
- Surface integral ex2 part 2
- Surface integral ex3 part 1
- Surface integral ex3 part 2
- Surface integral ex3 part 3
- Surface integral ex3 part 4

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Surface integral example, part 1

Visualizing a suitable parameterization. Created by Sal Khan.

## Want to join the conversation?

- what does it mean to not just integrate the surface but time it with x^2?(12 votes)
- In the videos before, Sal calculated the surface area. You can think of this as summing up the
*number*of the tiny surface elements – which is the same as assigning each surface element a value of one and then summing up over all ones.

In the current video, Sal assigns each surface element a different value, namely x^2, depending on the surface element's x position (he could have also chosen a value that depends on x, y, and z, but this makes the example more simple). You can think of the surface integral as summing up all the different*values*of all surface elements.(5 votes)

- At7:42, why does he multiply cos(t) with cos(s)? I don't understand the reasoning behind this?(9 votes)
- Despite being a very old comment:

On the xy plane, cos(s) and sin(s) would give us the border of the circle of unit 1 (there's an implicit "1" multiplying "cos(s)" and "sin(s)"). But looking to the z plane (the left Sal's drawing at7:42), you can see that the vector of unit 1, that's making a "t" angle with the xy plane, projects a vector on the xy plane (and the size of that vector is the cos(t)). Then, this projection on the xy plane (cos(t)) turned out to be the radius of the circle on the xy plane. So now, instead of an implicit "1" multiplying "cos(s)" and "sin(s)", we use "cos(t)" as the radius.

And knowing this, generalizing for any sphere of "a" radius, besides the cos(t) we can just multiply each component of the parameterization by "a", like this:

r(s, t) = (a*cos(t)*sin(s), a*cos(t)*cos(s), a*sin(t))(7 votes)

- @ https://www.youtube.com/watch?v=E_Hwhp74Rhc#t=583 , why doesnt the angle not need to go in the "other dirction as sal says" but is only constrained @ -pi/2 and pi/2?(5 votes)
- Sal didn't get this wrong. The way he set this up works. Here he has made the t kind of like a latitude coordinate. The s is set up like a longitude coordinate. So for t, it need only give how much north or south the point is above or below the equator. So using the standard orientation with N at the point (0,0,1), we have t is pi/2, and so z = sin t = sin (pi/2) = 1, and likewise at S = (0,0,-1), where t = -pi/2, and z = sin t = sin (-pi/2) = -1.(3 votes)

- Is it possible to solve the surface integral of a sphere without first parameterizing it?(3 votes)
- If the portion of the sphere you're integrating over is a function (like a hemisphere), then yes; you can use a geometric shortcut or the formula for functions (not recommended).

If it's the whole sphere, then you should parametrize it in spherical coordinates. The surface area element is related to the Jacobian (I can't remember exactly how this instant though), so you don't have to calculate the entire cross product.

You may also be able to use Gauss's Divergence Theorem, though then you'll probably end up switching to spherical coordinates anyway.(5 votes)

- Is it possible to integrate over 1dσ?(4 votes)
- my initial sentiment exactly. what's the intuition or purpose of multiplying by x^2 ?(3 votes)

- It seems that Sal comes up with a strategy on the fly to parameterize a surface with two variables. Is there a method that can be used that works every time or nearly every time, that can give you a parameterization?(3 votes)
- You could actually "parametrize" this surface by

x=s

y=t

z=f(x,y) = f(s,t).

You'd have to divide the surface into a top part and a bottom part for positive and negative z values, so that z is a function of x and y.

In that case, you probably wouldn't even bother changing the names of the variables - you'd just stick with calling them x and y, and you'd wind up integrating over dxdy. And it would work, and it is perfectly fine to do it that way, and in some cases it works out nicely. In some cases it is the best "parametrization", even if it seems too simple to bother calling it that.

But if you try to do this particular example that way, it gets messy with square roots all over the place.

Using the geometry of the surface to choose the parametrization leads to simpler math. That's the main purpose. But just going with a parametrization like x=x and y=y and z=f(x,y) does work.(4 votes)

- Curious, will I get a different answer if evaluating the surface integral with a different parameterization?(2 votes)
- No. Even if you used different parameterizations, they would give you the same surface integral, as long as those parameterizations were consistent with the surface.(3 votes)

- At5:35, Sal says that the radius will be smaller than it was before, but I'm confused because I thought that the radius of a unit sphere is always constant. Does he mean the z-coordinate or something like that instead of the radius?(3 votes)
- think in 3D... take another plane // to xy plane... and it intersects the sphere above the origin...

it is going to form a smaller circle, and obviously its radius is going to be smaller than unity.(1 vote)

- I don't get the intuition behind letting t range from -pi/2 to pi/2.

Shouldn't it range from 0 to 2pi?(3 votes) - At5:15, he mentions that "taking the cross section that is parallel to the xy-plane will give a smaller radius" what's the point of taking the cross section? what's the point of getting a smaller radius? If it's a sphere, shouldn't the radius always be the same?(2 votes)
- We have t being used to determine x and y, so we need something else to determine z, so that is why we take different cross sections at different values of z. The cross section is getting a smaller radius because some of the radius is being used for the z-component, so not as much can be used for the x and y components.(2 votes)

## Video transcript

What we will attempt to start
to do in this video is take the surface integral
of the function x squared over
our surface, where the surface in question,
the surface we're going to care about is
going to be the unit sphere. So it could be
defined by x squared plus y squared plus z
squared is equal to 1. And what I'm going to focus on
in this first video, because it will take us several
videos to do it, is just the parameterization of
this surface right over here. And as you'll see, this
is often the hardest part because it takes a little
bit of visualization. And then after that,
it's kind of mechanical, but it can be kind of
hairy at the same time. So it's worth going through. So first, let's think about
how we can parameterize-- and I have trouble
even saying the word. How we can parameterize
this unit sphere as a function of two parameters. So let's think about it. Let's think about
it a little bit. So first, let's just think
about the unit sphere. I'm going to take a side
view of the unit sphere. So let's take the unit sphere. So this right over
here is our z-axis. That's our z-axis. And then over here,
I'm going to draw-- this is going to be not
just the x or the y-axis. This is going to be entire
xy-plane viewed from the side. That is the xy-plane. Now, our sphere,
our unit sphere, might look something like this. The unit sphere itself is
not too hard to visualize. It might look
something like that. The radius-- let me
make it very clear. The radius at any point is 1. So this length right
over here is 1. That length right
over there is 1. And this is a sphere,
not just a circle. So I could even shade
it in a little bit, just to make it clear
that this thing has some dimensionality to it. So that's shading it in. It kind of makes it look a
little bit more spherical. Now, let's attempt
to parameterize this. And as a first step,
let's just think. If we didn't have to think
above and below the xy-plane, if we just thought about where
this unit sphere intersected the xy-plane, how we
could parameterize that. So let's just think about it. So where it intersects
the xy-plane. It intersects it there, and
there, and actually everywhere. So it intersects it
right over there. So let's just draw
the xy-plane and think about that intersection,
and then we could think about
what happens as we go above and below the xy-plane. So on the xy-plane, this little
region where we just shaded in. So let me draw. So now you could view
this as almost a top view. The z-axis is now going to be
pointing straight out at you, straight out of the screen. So that's x. So let me draw it. So that's x, and then this
right over there is y. So this thing that we
were viewing sideways, now we're viewing
it from the top. And so now our unit
sphere is going to look something like
this viewed from above. What I just drew, this dotted
circle right over here, this is going to be where
our unit sphere intersects-- I labeled that y. That should be x. Don't want to
confuse you already. Let me clear that. So this is our x-axis. This is our x-axis. So this little
dotted blue circle, this is where our unit sphere
intersects the xy-plane. And so using this,
we can start to think about how to parameterize
at least our x- and y-values, our x- and
y-coordinates, as a function of a first parameter. So the first
parameter, we can think of something that is-- so this
is the z-axis popping straight out at us. So we're essentially,
if we're rotating around that z-axis viewed from above,
we could imagine an angle. I will call that angle
s, which is essentially saying how much we're
rotating from the x-axis towards the y-axis. You could think about
it in the xy-plane or in a plane that is
parallel to the xy-plane. Or you could say, going
around the z-axis. The z-axis popping
straight up at us. And the radius here is always 1. It's a unit sphere. So given this
parameter s, what would be your x- and y-coordinates? And now we're thinking
about it right if we're sitting
in the xy-plane. Well, the x-coordinate--
this goes back to the unit circle definition
of our trig functions. The x-coordinate is
going to be cosine of s. It would be the radius, which
is 1, times the cosine of s. And the y-coordinate would
be 1 times the sine of s. That's actually where we get our
definitions for cosine and sine from. So that's pretty
straightforward. And in this case, z is
obviously equal to 0. So if we wanted to add our
z-coordinate here, z is 0. We are sitting in the xy-plane. But now, let's think
about what happens if we go above and
below the xy-plane. Remember, this is
in any plane that is parallel to the xy-plane. This is saying how we are
rotated around the z-axis. Now, let's think about if
we go above and below it. And to figure out how
far above or below it, I'm going to introduce
another parameter. And this new parameter I'm
going to introduce is t. t is how much we've rotated
above and below the xy-plane. Now, what's
interesting about that is if we take any other cross
section that is parallel to the xy-plane now, we are
going to have a smaller radius. Let me make that clear. So if we're right over there,
now where this plane intersects our unit sphere, the
radius is smaller. The radius is smaller
than it was before. Well, what would
be this new radius? Well, a little bit
of trigonometry. It's the same as this
length right over here, which is going to
be cosine of t. So the radius is going
to be cosine of t. And it still works
over here because if t goes all the way to
0, cosine of 0 is 1. And then that works
right over there when we're in the xy-plane. So the radius over here
is going to be-- so that right over
there is cosine of 0. So this is when t is equal to 0. And we haven't rotated
above or below the xy-plane. But if we have rotated
above the xy-plane, the radius has changed. It is now cosine of t. And now we can use that to truly
parameterize x and y anywhere. So now, let's look at
this cross section. So we're not necessarily
in the xy-plane, we're in something that's
parallel to the xy-plane. And so if we're up here,
now all of a sudden, the cross section-- if
we view it from above, might look something like this. It might look
something like this. We're viewing it from
above, this cross section right over here. Our radius right over
here is cosine of t. And so given that--
I guess altitude that we're at, what would now
be the parameterization using s of x and y? Well, it's the exact same
thing, except now our radius isn't a fixed 1. It is now a function of t. So we're now a
little bit higher. So now, our
x-coordinate is going to be our radius,
which is cosine of t. That's just our radius. Times cosine of s. Times cosine of s, how
much we've angled around. And in this case, s has gone
all the way around here. S has gone all the
way around there, so it's going to be cosine
of t times cosine of s. And then, our
y-coordinate is going to be our radius, which is
cosine of t times sine of s. Same exact logic here, except
now we have a different radius. Our radius is no longer 1. Times sine of s. I'm running out of space,
let me scroll to the right a little bit. And I know this
looks very confusing, but you just have to say,
at any given level we are, we're parallel to the x-axis. We're kind of tracing
out another circle where another plane intersects
our unit sphere. We're now then
rotating around with s. And so our radius will change. It's a function of how
much above and below we've rotated-- how much
above or below the xy-plane we've rotated. So this is just our
radius instead of 1. And then, s is how much we've
rotated around the z-axis. Same there for the y-coordinate. And then the z-coordinate
is pretty straightforward. It's going to be
completely a function of t. It's not dependent on how
much we've rotated around here at any given altitude. It is what our
altitude actually is. Now, we can go straight to
this diagram right over here. Our z-coordinate is just
going to be the sine of t. So our z is equal to sine of t. So let me write that down. So the z is going to
be equal to sine of t. So now, every point
on this sphere can be described as a
function of t and s. Now, we have to think
about over what range will they be defined. Well, s is going to go-- at any
given level, you could think. For any given t, s is going
to go all the way around. We see that right over here. At any given level
viewed from above, s is going to go
all the way around. So thinking about
in radians, s is going to be between 0 and 2 pi. And t is essentially our
altitude in the z-direction. So t can go all
the way down here, which would be
negative pi over 2. So t can be between
negative pi over 2. And it can go all the
way up to pi over 2. It doesn't need to go all
the way back down again. And so it goes all the way
back-- it goes only up to pi over 2. And then we have our
parameterization. Let me write this down
in a form that you might recognize even more. If we wanted to write our
surface as a position vector function, we could
write it like this. We could write it r is
a function of s and t, and it is equal to
our x-component. Our i-component is going to
be cosine of t cosine of s i. And then plus our y-component
is cosine of t sine of s plus our z-component,
which is the sine-- which is just-- oh, I
forgot our j vector. j plus the z-component, which
is just sine of t sine k. And we're done. And these are the ranges that
those parameters will take on. So that's just the first step. We've parameterized
this surface. Now we're going to
have to actually set up the surface integral. It's going to involve
a little bit of taking a cross product,
which can get hairy, and then we can actually
evaluate the integral itself.