Plain area

Illustration of a normal area with regard to x

A plain area , normal area or normal area is a mathematical object from analysis . It is a for calculus easy-to-use area .

definition

It is defined as a plain area

${\displaystyle G=\lbrace (x,y)\in \mathbb {R} ^{2}\;|\;a<x<b,\;g(x)<y<h(x)\rbrace }$

with ( continuous) and simply means above the -axis or A) is the normal range with respect to . If the limits of fixed and the limits of continuous functions are above , then the plain area is simply called above the -axis or B) is normal area with respect to . If properties A) and B) apply, this is called a normal range . The same applies in that each coordinate is considered once within fixed limits and all other coordinate limits as a graph of continuous functions. ${\displaystyle g,h\colon [a,b]\longrightarrow \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle y}$ ${\displaystyle G}$ ${\displaystyle y}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \mathbb {R} ^{n}}$

Explanation

It is a subset of the real number level. To determine it, one needs two real numbers and two continuous functions provided by the closed interval to represent. ${\displaystyle a,b}$ ${\displaystyle g,h}$ ${\displaystyle [a,b]}$ ${\displaystyle \mathbb {R} ^{1}}$

Then the plain domain is defined as those pairs of numbers for which: ${\displaystyle (x,y)}$

${\displaystyle a<x<b}$ and ${\displaystyle g(x)<y<h(x).}$

surface

Usually one of the first uses of calculus in schools is to calculate the area of ​​a plain area. The area of ​​the area is calculated using the following integral: ${\displaystyle G=\lbrace (x,y)\in \mathbb {R} ^{2}\;|\;a<x<b,\;g(x)<y<h(x)\rbrace }$

${\displaystyle \int _{a}^{b}(h(x)-g(x))dx}$.

More complicated areas are then often composed of simple areas.

literature

• Harro Heuser, Textbook of Analysis Part 2 , Teubner, Stuttgart, 1992, ISBN 3-519-12232-4