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## Multivariable calculus

### Course: Multivariable calculus > Unit 4

Lesson 12: Surface integrals (articles)# Surface integrals

How do you add up infinitely many infinitely small quantities associated with points on a surface?

## Background

Not strictly required, but useful for intuition and analogy:

## What we're building to

- In principle, the idea of a surface integral is the same as that of a double integral, except that instead of "adding up" points in a flat two-dimensional region, you are adding up points on a surface in space, which is potentially curved. The abstract notation for surface integrals looks very similar to that of a double integral:

- Computing a surface integral is almost identical to computing surface area using a double integral, except that you stick a function inside the integral:

Here, $\overrightarrow{\mathbf{\text{v}}}({t},{s})$ is a function parameterizing the surface $S$ from the region $T$ of the ${t}{s}$ -plane.

(This is analogous to how computing line integrals is basically the same as computing arc length integrals, except that you throw a function inside the integral itself.)

- You can find an example of working through one of these integrals in the next article.

## The idea of surface integrals

If you understand double integrals, and you understand how to compute the surface area of a parametric surface, you basically already understand surface integrals. It's just a matter of smooshing the two intuitions together. I'll go over the computation of a surface integral with an example in just a bit, but first, I think it's important for you to have a good grasp on what exactly a surface integral

*does*.#### Refresher of double integrals

Recall what a double integral does:

Here, $R$ represents some region of the $xy$ -plane, and $f(x,y)$ is a way to associate each point of $R$ with a number.

- Maybe
represents a metal sheet, and$R$ represents the density at each point.$f(x,y)$ - Or perhaps
represents a geographic region, and$R$ represents the temperature at each point.$f(x,y)$

The double integral provides a way to "add up" the values of $f$ on this region. However, the idea of "adding up" points in a continuous region is vague, so I like to imagine the following process:

- Chop up the region
into many tiny pieces.$R$ - Multiply the area of each piece, thought of as
, by the value of$dA$ at one of the points inside that piece.$f$ - Add up the resulting values.

For example,

- If
represents a metal sheet, and$R$ is a density function, the double integral will give you the mass of the sheet. (Why?)$f(x,y)$ - If
represents a geographic region, and$R$ give the temperature at each location, taking this double integral then dividing by the area of$f(x,y)$ will give you the average temperature in that region. (Why?)$R$

#### Double integrals over curved regions

However, why stay so flat? This idea of adding up values over a continuous two-dimensional region can be useful for curved surfaces as well.

- What if you are considering the surface of a curved airplane wing with variable density, and you want to find its total mass?
- What if you have the temperature for every point on the curved surface of the earth, and you want to figure out the average temperature?

This time, the function $f$ , which represents density, temperature, etc., must take in point of three dimensions since points on the surface live in three dimensions. The abstract notation for integrating a three-variable function $f(x,y,z)$ over a surface is pretty much the same as the abstract notation for double integrals:

(Different authors might use different notation).

This is called a $S$ under the double integral sign represents the surface itself, and the term $d\mathrm{\Sigma}$ represents a tiny bit of area piece of this surface. You can think about surface integrals the same way you think about double integrals:

**surface integral**. The little- Chop up the surface
into many small pieces.$S$ - Multiply the area of each tiny piece by the value of the function
on one of the points in that piece.$f$ - Add up those values.

Why write $d\mathrm{\Sigma}$ instead of $dA$ ? There's no real difference; each one represents a tiny bit of area of the thing you are integrating over. However, when it comes time to compute things, the way to handle a tiny bit of area on a curved surface is fundamentally different from doing it on a flat surface, so it's worth emphasizing this difference by using a different variable.

## How to compute a surface integral

Abstract notation and visions of chopping up airplane wings are all well and good, but how do you actually

*compute*one of these surface integrals? The trick is to sneakily turn it into an ordinary,*flat*, double integral.Specifically, the way you tend to represent a surface mathematically is with a parametric function. You'll have some vector-valued function $\overrightarrow{\mathbf{\text{v}}}(t,s)$ , which takes in points on the two-dimensional $ts$ -plane (lovely and flat), and outputs points in three-dimensional space. You also need to specify the region $T$ of the $ts$ -plane which maps onto the surface $S$ .

The trick for surface integrals, then, is to find a way of integrating over the flat region $T$ that gives the same effect as integrating over the curved surface $S$ . This requires describing "tiny piece of area" of $S$ in terms of something inside the parameter.

Almost all of the work for this was done in the article on surface area. There, we saw how a tiny rectangle inside $T$ with area ${dt}{\textstyle \phantom{\rule{0.167em}{0ex}}}{ds}$ gets transformed into a parallelogram on $S$ with area
$\begin{array}{r}|{\displaystyle \frac{\partial \overrightarrow{\mathbf{\text{v}}}}{\partial {t}}}\times {\displaystyle \frac{\partial \overrightarrow{\mathbf{\text{v}}}}{\partial {s}}}|{\textstyle \phantom{\rule{0.167em}{0ex}}}{dt}{\textstyle \phantom{\rule{0.167em}{0ex}}}{ds}\end{array}$

For our surface integral desires, this means you expand $d\mathrm{\Sigma}$ as follows:

Specifically, here's how to write a surface integral with respect to the parameter space:

Let's break that down a bit:

The main thing to focus on here, and what makes computations particularly labor intensive, is the way to express $d\mathrm{\Sigma}$ .

In the next article, you can go through a full example of one of these surface integrals.

## Summary

- Surface integrals are used anytime you get the sensation of wanting to add a bunch of values associated with points on a surface. This is the two-dimensional analog of line integrals. Alternatively, you can view it as a way of generalizing double integrals to curved surfaces.

- Computing a surface integral is almost identical to computing surface area using a double integral, except that you stick a function inside the integral:

Like so many things in multivariable calculus, while the theory behind surface integrals is beautiful, actually computing one can be painfully labor intensive.

## Want to join the conversation?

- I almost went crazy over this but note that when you are looking for the SURFACE AREA (not surface integral) over some scalar field (z = f(x, y)), meaning that the vector V(x, y) of which you take the cross-product of becomes V(x, y) = (x, y, f(x, y)).

This also means that the cross-product is evaluated too:

sqrt( (partial-x-of-f(x,y))² + (partial-y-of-f(x,y))² + 1 )

Since we are looking for the area and not the integral, the equation then becomes:

double-integral-over-S (dS) =

double-integral-over-S (sqrt( (partial-x-of-f(x,y))² + (partial-y-of-f(x,y))² + 1 ) dxdy )

Better explanation: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx(6 votes) - Why do you add a function to the integral of surface integrals?(1 vote)
- Well because surface integrals can be used for much more than just computing surface areas. It's like with triple integrals, how you use them for volume computations a lot, but in their full glory they can associate any function with a 3-d region, not just the function f(x,y,z)=1, which is how the volume computation ends up going. Double integrals also can compute volume, but if you let f(x,y)=1, then double integrals boil down to the capabilities of a plain single-variable definite integral (which can compute areas). Having an integrand allows for more possibilities with what the integral can do for you.(5 votes)

- What about surface integrals over a vector field?(2 votes)
- That is known as flux(1 vote)

- So, this is essentially a surface area integral when considering a function f(x,y,z) instead of a vector-valued function?(1 vote)
- Wow thanks guys! I understood this even though I'm just a senior at high school and I haven't read the background material on double integrals or even Calc II(0 votes)
- Wow what you're crazy smart how do you get this without any of that background?(1 vote)