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Homesurface integration with respect to area

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# surface integration with respect to area

This entry is still in the process of being written and, since it is rather lengthy, it may take some time to complete.

# 1 Introduction

In pure and applied math, one frequently encounters integrals over surfaces with respect to surface area. The simplest instance occurs when computing the area of a surface. More complicated instances occur when one is computing, say the moments of a shell or the total force acting on a surface. In this entry, methods of computing such integrals together with their theoretical justifications will be discussed.

The material to be found in this entry varies greatly in the level of mathematical sophistication. At the low end, there are formulas for computing such integrals which can be understood and used profitably by a calculus student. At the high end, a deeper discussion of this topic requires such relatively advanced mathematical tools as exterior calculus, Riemannian geometry, and measure theory. On the one hand, because this is an encyclopaedia, the coverage should be as thorough as possible. On the other hand, discussing the material at a high level of mathematical sophistication would make it inaccessible to many readers who could benefit from it. As a way out of this dilemma, we have chosen to write this entry in several sections with the level of mathematical sophistication and rigor incerasing from section to section. In the beginning, we will only assume familiarity with integral calculus and derive formulas using heuristic arguments. Rigorous proofs will only come in later sections because they require mathematical tools which may not yet be available to the typical calculus student. Likewise, in the earlier sections we shall treat the three-dimensional case separately from the more general formailsm that applies in any number of dimensions.

# 2 Calculus

# 2.1 Basic idea and definition

Let $S$ be a surface in three-dimensional space ($\mathbb{R}^{3}$) and let $f$ be a function defined on this surface. To describe the surface, we will choose a parameterization of the surface by two variables which we shall call $u$ and $v$. (For instance, if $S$ is a sphere, $u$ and $v$ might be the latitude and longitude.) In terms of this parameterization, the function $f$ can be expressed as a function of $u$ and $v$. Then the *integral of $f$ with respect to the surface area* is usually notated as

$\int_{S}f(u,v)\,d^{2}A.$ |

Just as with the ordinary integral, this quantity may be understood as a limit of sums.^{1}^{1}Note on rigour, or lack thereof:
For our definition of integral to be sound, we need to show that the limit exists and depends only on the surface $S$ and the function $f$ and not on the details of how we choose to subdivide the surface. Throughout this section, we shall sweep such questions of mathematical propriety under the rug for three reasons: 1) In this section, our main interest is in deriving practical formulas and procedures which are of use in computing surface integrals that arise in practise. 2) This section is written for the benefit of beginners who may not be familiar with the techniques necessary to properly justify the manipulations presented here; the more sophisticated reader may skim through the formulas and examples and proceed to the later sections. 3) “The physicist’s excuse” — As long as we restrict ourselves to rather familiar and well-behaved surfaces and functions, our intuition should save us from making really serious mistakes. (Along the same lines, it is worth pointing out that we shall also adopt the naive point of view that differentials like $dx$ represent tiny displacements rather than a more sophisticated interpretation such as differential forms.) To form an approximating sum, we subdivide the surface $S$ into a miniscule pieces, multiply the area of each piece by the value of $f$ at a point located inside that piece, sum over all the miniscule pieces into which we have subdivided the surface. The limiting value of these sums as the size of the pieces shrinks to zero is the integral with respect to surface area.

To compute this quantity, one may use the following formula to convert it to a double integral:

$\int_{S}f(u,v)\,d^{2}A=\int f(u,v)\sqrt{\left(\frac{\partial(x,y)}{\partial(u,% v)}\right)^{2}+\left(\frac{\partial(y,z)}{\partial(u,v)}\right)^{2}+\left(% \frac{\partial(z,x)}{\partial(u,v)}\right)^{2}}\>du\,dv.$ |

If the surface is described as the graph of a function $g$, then we may also use the following formula:

$\int_{S}f(x,y)d^{2}A=\int f(x,y)\sqrt{1+\left(\frac{\partial g}{\partial x}% \right)^{2}+\left(\frac{\partial g}{\partial y}\right)^{2}}\>dx\,dy.$ |

# 2.2 Examples

To explain the use of these formulae, several worked examples have been presented in the forms of supplements to this entry. The first four examples illustrate the formula for integrals over parameterized surfaces and the latter four examples deal with surfaces presened as graphs of functions.

Example 1. This example shows how integrals over spheres with respect to surface area may be rewriten as integrals with respect to the spherical coordinates.

Example 2. This example builds on the previous example by carrying out the computation of an integral over the sphere using the formula derived in example 1.

Example 3. This example shows how to re-express integrals over helicoids as integrals over the parameters.

Example 4. This example uses the result of example 3 to compute the area of a portion of a helicoid.

Example 5. In this example, we revisit the sphere and consider it as the graph of the function $g(x,y)=\sqrt{x^{2}+y^{2}}$.

Example 6. We use the result of example 5 to compute the moment of inertia of a spherical shell.

Example 7. In this example, we consider integration on the paraboloid described by the equation $z=x^{2}+3y^{2}$.

Example 8. We use the result of example 7 to compute the area of a portion of the paraboloid. From a purely technical point of view, this example is the longest and most complicated of the examples. Evaluating the integral is a long process and the answer involves elliptic integrals.

# 2.3 A common mistake

Before proceeding further, a caveat may be in order. Beginners are often apt to make the mistake that

$\int_{S}f(u,v)\,d^{2}A=\int_{S}f(u,v)\,du\,dv.$ |

By a fluke, this may give the right answer in certain cases, but as a general principle, IT IS WRONG. (This is a lot like the fact that in elementary arithmetic, one cannot reduce a fraction to lowest terms by striking out digits that appear in both the numerator and denominator, even if this happens to give the right answer in a few cases.) Since this mistake is so common and easy to make, it might not be inappropriate if we take out some time to explain why the above formula is wrong. The reason is that $d^{2}A$ refers to the area of a small portion of surface whilst $du\,dv$ refers to the change in $u$ multiplied by the change in $v$. In general, these two quantity are not equal for two reasons.

First, to compute an area, we need to multiply two lengths, but $u$ and $v$ may not always measure length. The following example should help clarify this problem. Suppose that $S$ is a sphere and that the parameters $u$ and $v$ are latitude and longitude, respectively. Then, as every navigator knows, if we travel due west a degree of longitude starting in Florida ($dv=1$), we will have travelled a greater distance than if we travel due west a degree of longitude starting in Sweden (also $dv=1$, but different values for $u$ and $v$). In fact, to obtain the distance travelled, we must multiply the change in longitude by the cosine of the latitude. In fact, this is why, as we shall see, on a unit sphere the correct formula is

$d^{2}A=\cos u\,du\,dv,$ |

Second, even when $u$ and $v$ measure length, they may not be perpendicular. If so, then the portion of surface which is traced out by letting the one parameter vary between $u$ and $u+du$ and letting the other parameter vary between $v$ and $v+dv$ will look more like a parallelogram than a rectangle. As every student of plane geometry knows, the product of the length of the sides will not equal the area of the parallelogram except in the special case when it happens to be a rectangle.

# 2.4 Derivation of formulas for area integration

Having discoursed at some length about what not to do, let us now derive a correct formula for $d^{2}A$. We shall show how to do this in two different ways, not only for the sake of completeness, but because one method is better adapted to deriving one formula and the other method is better adapted to deriving the other formula.

Also, it is worth checking explicitly that the answer depends only on the surface $S$ and the function $f$ but not on the choice of parameterization. This check is a routine application of the chain rule and the rule for change of variables in multiple integrals.

## Mathematics Subject Classification

28A75*no label found*

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## Attached Articles

example of integration with respect to surface area on a helicoid by rspuzio

integration with respect to surface area on a helicoid by rspuzio

example of integration with respect to surface area on a sphere viewed as a graph by rspuzio

computation of moment of spherical shell by rspuzio

example of integration with respect to surface area of a paraboloid by yark

computation of surface area of portion of paraboloid by rspuzio

derivation of first formula for surface integration with respect to area by rspuzio

derivation of second formula for surface integration with respect to area by rspuzio

invariance of formula for surface integration with respect to area under change of variables by rspuzio

## Comments

## integrate <over> or <on> ?

Does anybody know what is the right preposition for the verb 'integrate', i.e. which of the following phrases is right:

function f(x) is integrated <over> a sphere

function f(x) is integrated <on> a sphere

or may be they both are correct?

## Re: integrate <over> or <on> ?

Intereseting question. While they may not be of earth-shattering importance (and certainly are mathematically irrelevant), linguistic questions like this can be fun and interesting diversions, so I decided to have a look. After the business with the tuplets, I already had the dictionary open and ready for business.

I decided to start by browsing through a few math books to see what usages I might find. Unfortunately, this can be tricky because, as often as not, someone simply writes an expression like \int_S and doesn't bother describe what they are doing using words.

Generally speaking, I saw the phrase "intgration over" used most frequently. However, other prepositions also showed up. For instance, Whittaker and Watson use the phrase "integration along" in their discussion of Cauchy's theorem.

My guess would be that both usages are acceptable. From the syntactic point of view, "on" and "over" are prepositions, so both usages parse correctly. The question is one of semantics. However, given a well-behaved function and a sphere, there's a unique number which can be rightly called the integral of that function, so no matter which phrase one uses, it will be understood to refer to the same quantity. I suppose that if one where dealing with pathological functions for which one could come up with different anwers by using different procedures, then one might decide to define "integration on" to refer to one procedure and "integration over" to refer to the other. As far as I know, noone has done this.

Personally, I prefer the term "over" and would usually say something like "given a function f on the sphere, we integrate this function f over the sphere with respect to the measure m on the sphere". I guess that this is because, for me, the word "on" in mathematics suggests a structure (like a function or a measure) that just sits on the sphere and looks at you waiting for something to happen while "over" suggests a process that is happening and moves from one part of the sphere to another. This is kind of like what happens in everyday language where one speaks of "standing ON a rock" as opposed to "walking OVER a rock".

Maybe the difference is one of connotation --- "integrate over the sphere" suggests that to me that the integral is a process that involves the sphere whilst "integrate on" suggests that the sphere is a passive bystander. Thinking this way, the choice of word would be a reflection of one's intuitive picture of integration.

Since we have an active Finnish-speaking member on this site, it might be interesting to compare this with what happens in Finnish, since that language belongs to a different family.

P.S. For the heck of it, I had a look at the Oxford English dictionary and found that they allowed -tuple as a noun, so I guess both usages are OK. In this case, listing both as synonyms should solve the problem.

## Re: integrate <over> or <on> ?

in spanish both on and over can be described with the same word "sobre" and thus we say

"la integral sobre una esfera|region, etc"

I don't think "along" is used other than over curves, in spanish we use "a lo largo de" for along but it's also heard

"integral sobre la curva"

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: integrate <over> or <on> ?

Indeed the Finnish language is not Indo-European but a Uralic (or Finno-Ugrian) language, the predecessor of which was spoken as a lingua franca under the last Ice Age widely in the northern Central and Eastern Europe (in the Periglacial Zone); nowadays the Uralic languages form only separate insels (as Estonian, Hungarian, Mordwinian, Cheremis, Nenetsian etc.). Many scientists say that e.g. the Germans have changed their original Finno-Ugric language to an Indo-European one, when the Indo-Europeans came from the south bringing the agriculture to those mammouth-hunters (in the German one can find some Finno-Ugric substrate features).

As for integration over some domain, we use in Finnish the comparable preposition or postposition "yli" -- I think this usage has come to us from other European languages (as from French "sur", German "Ã¼ber" or Swedish "Ã¶ver"). In the line integrals we use the pre- or postposition "pitkin" which corresponds the English word "along".

## Re: integrate <over> or <on> ?

While we're on the subject, it might be worth mentioning that in Polish, the usual expression for "intgrate over a sphere" is "calkowac po kule", although Banach prefers "calkowac w kule" --- the preposition "po" in this context (this word tends to be used with many different meanings) can be translated as "over" (or as "on") and "w" as "in".

What I find interesting about linguistics and mathematics is how the meanings of words get stretched from their usual meanings in order to describe abstract concepts which can be very far removed from the everyday experiences which language was designed for. A good example would be the phrase "closed set" --- clearly the sense of the word "closed" has changed from the sense in a phrase like "closed door".

This discussion again brings up the suggestion that has been made on more than one occasion in the past of adding some sort of lexicon to Planet Math. I'm wondering how hard or easy it would be to add a place in the entries for foreign terms. There's already a place for pronounciation; I assume that this could be something similar.

## Re: integrate <over> or <on> ?

Many thanks for replies! The main reason for the question was that there is to my opinion inconsistency in the following names:

"example of integration with respect to surface area on a helicoid"

and

"example of integration with respect to surface area of the paraboloid"

so taking into account all discussions they both should be changed to:

"example of integration with respect to surface area OVER a helicoid"

"example of integration with respect to surface area OVER the paraboloid"

also what seems strange is once usage of indefenite article and other times usage of definite one. Probably one should only one, but which?

--------------------------

What also seems quite strange to me is that googling the expression

"integral with respect to surface area"/"integrals with respect to surface area"

brings very few results, whereas

"surface integral"/"surface integrals"

seem to be much more popular. Thus may be the better name for the entry

"integration with respect to surface area"

would be just

"surface integration"

which is much shorter and easier readable, or? At least to make it in 'other name' field won't hurt. But for the names of examples I would choose "surface integration" since it would be defenitely much more readable

"example of surface integration over a helicoid"

(it seems that defenite article is more appropriate) then current

"example of integration with respect to surface area on a helicoid".

Objections?

## Re: integrate <over> or <on> ?

To be consistent, I will change all ocurences of any other preposition to "over" and revise the names of the titles. I am still thinking about your suggestion to consolidate some of these entries. I am also thinking of other ideas such as attaching a short table of formulas for surface integrals over several popular surfaces (e.g. spheres, cones, cylinders, ellopsiods, ...) for as a handy refernce. In this case maybe I would turn some of these examples into derivations of the formulas in that table. All in all, this entry is in flux as I experiment with different ways of presenting this material and organizing it.

I deliberately chose the title "integration with respect to surface area" rather than "surface integrals" for a specific technical reason. Whilst the term "surface integration" denotes any kind of integral over a surface, the phrase "with respect to surface area" specifies that one is using a specific measure to perform the integral. Since the focus of this article is on computing integrals with respect to this measure, I thought this title would be more appropriate. Perhaps a better choice of title would be "surface integrals with respect to area measure", but I feel reluctant to use the term "measure" because this entry is meant to be accessible to calculus students and one usually doesn't hear about measures until one get to analysis class. As a compromise, I'll try the title "surface integrals with respect to area". If that's not too good either, let me know and I'll try again.

However, adding "surface integration" as a related topic or a keyword is an excellent idea, so I will do that.

In closing, I would like to thank Serg and Robert for their suggestions with respect to organizing this entry. In many ways, writing a Planet Math entry differs from more conventional mathematical writng, such as writing a book or an article. One runs into issues like using hypertext effectively and keeping terminology and notation consistent with other entries. One must keep in mind that an entry might be accessed by readers with different goals and levels of mathematical sophistication and that the links should be understandable to sthe reader. While it is relatively straightforward to write short entries, writing, editing and organizing a longer entry like this one can be something of a learning experience. Your advice and suggestions are most helpful and have already led to an improved entry.

Thanks,

Ray

## Re: integrate <over> or <on> ?

Dear Ray

First of all many thanks for your kind words:

> In closing, I would like to thank Serg and Robert

> for their suggestions with respect to organizing this entry.

> Your advice and suggestions are most helpful

> and have already led to an improved entry.

What you wrote about difficulties in writing encyclopedia entries is completely true (at least I feel that way), but you seem succesfully overcome them, when looking on the amount of materials you are producing ;). It is really amazing how fast you can write the texts ;).

Now back to our discussion:

> Whilst the term "surface integration"

> denotes any kind of integral over a surface

In fact, I havn't seen any other integrations over the surface except the one you are taling about in the entry. But it is probably possible to have some disastrous measures on the surface and thus having some strange integrals. Thus I would probably keep your old title "..." (you have already changed it so I can't type it :). The reason I find your new title:

"surface integration with respect to area"

suspicuous, is that if you google it, then you'll see nothing (probably after some time it will give your entry), which means that such term is extremely rare in use. By the way the same was with your old title, and that's why I proposed "surface integration" as more frequently used term. Thus, I would choose some name used in most books on calculus.

Anyway, as you said this entry is in flux, thus we'll keep on talking about improvements.

Regards

Serg.

## Re: integrate <over> or <on> ?

Interesting question indeed. I think "over" is preferable -- i guess i think of integration as a process that sweeps across the surface, so to say. Both can be defended, of course.

Come to think of it, "on" is wider in its applicability. If you draw a curve on the sphere, and do a line integral along the curve, you're still integrating "on the sphere" (as opposed to a line integral on a plane say). Integrating "over" the sphere only feels right for \int something d^2 A, but it too is still an instance of doing your integration "on" the sphere.

On a related note: i'm never sure whether to say a graph is embedded "on" or "in" a sphere. OT1H the "em-" in "embedded" really seems to want "in". OTOH i find myself using "on" to emphasise it's a surface (maybe only to avoid some readers getting the wrong idea and thinking i'm talking about the interior of a solid ball).

Oh, before i forget: in the (perhaps aptly named ;) section "A common mistake": some typos

s/exmaple/example/

s/Sweeden/Sweden/

--regards, marijke

http://web.mat.bham.ac.uk/marijke/

## Re: integrate <over> or <on> ?

Thanks for putting a note on the old theme! ;)

> s/exmaple/example/

> s/Sweeden/Sweden/

For the above typos it is better to submit a correction to the entry ;)

Regards

SeRG.

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: integrate <over> or <on> ?

Don't bother with the corrction in this case, because I read the message and made the changes. However, as a general policy, a corection is an appropriate place to point out mispellings.