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A selection of theorems about integration in the plane. Theorems from other modules from the same author are referenced, and practice exercises are included.

Let S be a closed geometric set in the plane. If f is a real-valued function on S , we would like to define what it means for f to be “integrable” and then what the “integral” of f is. To do this, we will simply mimic our development for integration of functions on a closed interval [ a , b ] .

So, what should be a “step function” in this context? That is, what should is a “partition” of S be in this context? Presumably a step function is going to be a function that is constant on the “elements” of a partition.Our idea is to replace the subintervals determined by a partition of the interval [ a , b ] by geometric subsets of the geometric set S .

The overlap of two geometric sets S 1 and S 2 is defined to be the interior ( S 1 S 2 ) 0 of their intersection. S 1 and S 2 are called nonoverlapping if this overlap ( S 1 S 2 ) 0 is the empty set.

A partition of a closed geometric set S in R 2 is a finite collection { S 1 , S 2 , ... , S n } of nonoverlapping closed geometric sets for which i = 1 n S i = S ; i.e., the union of the S i 's is all of the geometric set S .

The open subsets { S i 0 } are called the elements of the partition.

A step function on the closed geometric set S is a real-valued function h on S for which there exists a partition P = { S i } of S such that h ( z ) = a i for all z S i 0 ; i.e., h is constant on each element of the partition P .

REMARK One example of a partition of a geometric set, though not at all the most general kind, is the following.Suppose the geometric set S is determined by the interval [ a , b ] and the two bounding functions u and l . Let { x 0 < x 1 < ... < x n } be a partition of the interval [ a , b ] . We make a partition { S i } of S by constructing vertical lines at the points x i from l ( x i ) to u ( x i ) . Then S i is the geometric set determined by the interval [ x i - 1 , x i ] and the two bounding functions u i and l i that are the restrictions of u and l to the interval [ x i - 1 , x i ] .

A step function is constant on the open geometric sets that form the elements of some partition. We say nothing about the values of h on the “boundaries” of these geometric sets. For a step function h on an interval [ a , b ] , we do not worry about the finitely many values of h at the endpoints of the subintervals. However, in the plane, we are ignoring the values on the boundaries, which are infinite sets. As a consequence, a step function on a geometric set may very well have an infinite range,and may not even be a bounded function, unlike the case for a step function on an interval.The idea is that the boundaries of geometric sets are “negligible” sets as far as area is concerned, so that the values of a function on these boundaries shouldn't affect the integral (average value) of the function.

Before continuing our development of the integral of functions in the plane, we digress to present an analog of [link] to functions that are continuous on a closed geometric set.

Let f be a continuous real-valued function whose domain is a closed geometric set S . Then there exists a sequence { h n } of step functions on S that converges uniformly to f .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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