Површина — разлика између измена
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Ред 1:
[[Датотека:Area.svg|alt=Three shapes on a square grid|right|thumb|Ukupna površina ova tri oblika je približno 15,57 [[Kvadrat|kvadrata]].]]
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'''Површина''' је [[геометрија|геометријски]] појам који означава меру величине геометријске слике у еуклидском дводимензионалном простору. Тачка и линија немају површину, односно површина им је нула. Са друге стране [[раван]] има бесконачну површину. Површина је такође и део тела у простору који је изложен спољашњости. Мерењем површина су се бавили још стари Египћани, али су га до нивоа науке подигли тек [[античка Грчка|стари Хелени]]. Код њих се површина неке геометријске слике израчунавала тако што се низом трансформација претвара у [[квадрат]] исте површине. Потом се измере странице квадрата и лако израчуна површина.<ref name="AF">{{cite web|url = http://www.math.com/tables/geometry/areas.htm|title = Area Formulas|publisher = Math.com|accessdate = 2 July 2012}}</ref> Од тих дана је израчунавање површине добило други назив: ''квадратура''.
Површина је количина која описује у којој је мери дводимензионална фигура или облик, или планарне ламине, у [[Раван (математика)|равни]]. Површина је њен аналогни појам на дводимензионалној [[Површ|површи]] тродимензионалног облика. Површина може бити схваћена као количина материјала са датом дебљином која би била потребна да обуче модел облика, или количина боје потребне да прекрије површ при једном наносом.<ref name="MathWorld">{{cite web|url = http://mathworld.wolfram.com/Area.html|title = Area|publisher = [[Wolfram MathWorld]]|author = [[Eric W. Weisstein]]|accessdate = 3 July 2012}}</ref> То је дводимензионални аналог [[Дужина|дужине]] кривуље (једнодимензионални концепт) или [[Запремина|запремине]] чврстог тела (тродимензионални концепт).
У [[СИ систем|СИ систему]], стандардна јединица површине је [[квадратни метар]] (пише се као m<sup>2</sup>), што је површина квадрата чије су странице дуге по један [[метар]].<ref name="B">[[Међународни биро за тегове и мере|Bureau International des Poids et Mesures]] [http://www.bipm.org/en/CGPM/db/11/12/ Resolution 12 of the 11th meeting of the CGPM (1960)], retrieved 15 July 2012</ref> Облик са површином од три квадратна метра би имао исту површину као и три таква квадрата. У [[Математика|математици]], јединица квадрата је дефинисана да има површину од један, и површину од било којег облика или површи је [[Бездимензионална величина|бездимензиони реални број]].
Постоји неколико добро познатих формула за површине мањих облика као што су [[Троугао|троуглови]], [[Правоугаоник|правоугаоници]] и [[Кружница|кругови]]. Користећи ове формуле, површина сваког [[Полигон|полигона]] може се наћи дељењем полигона у троуглове.<ref name="bkos">{{Cite book |author1 = Mark de Berg|author2 = Marc van Kreveld|author3 = Mark Overmars|author3-link = Mark Overmars|author4 = Otfried Schwarzkopf|year = 2000|title = Computational Geometry|publisher = [[Springer-Verlag]]|edition = 2nd revised|isbn = 3-540-65620-0|chapter = Chapter 3: Polygon Triangulation|pages = 45–61 }}</ref> За облике са закривљеним границама, [[калкулус]] се често користи да се израчуна површина. Доиста, проблем одређивања површине равних фигура био је већа мотивација за историјски развој калкулуса (математичка анализа).<ref>{{cite book|first = Carl B.|last = Boyer|authorlink = Carl Benjamin Boyer|title = A History of the Calculus and Its Conceptual Development|publisher = Dover|year = 1959|isbn = 0-486-60509-4}}</ref>
За чврсти облик као што је [[сфера]], [[Купа (геометрија)|конус]] или цилиндар, површина њихових површи назива се површина површи.<ref name="MathWorld">{{cite web|url = http://mathworld.wolfram.com/Area.html|title = Area|publisher = [[Wolfram MathWorld]]|author = [[Eric W. Weisstein]]|accessdate = 3 July 2012}}</ref><ref name="MathWorldSurfaceArea">{{cite web|url = http://mathworld.wolfram.com/SurfaceArea.html|title = Surface Area|publisher = [[Wolfram MathWorld]]|author = [[Eric W. Weisstein]]|accessdate = 3 July 2012}}</ref> формуле за површине једноставних облика биле су рачунате у доба древних Грка, али рачунање површине компликованијих облика обично захтева мултиваријабилни калкулус.
Површина игра важну улогу у модерној математици. У додатку са очигледном важношћу у [[Геометрија|геометрији]] и калкулусу, површина је везана за дефиницију детерминанти у [[Линеарна алгебра|линеарној алгебри]], те је основна особина површи у диференцијалној геометрији.<ref name="doCarmo">[[Manfredo do Carmo|do Carmo, Manfredo]]. </ref> У [[Анализа|анализи]], површина подскупа равни је дефинисана кориштењем мере Лебега,<ref name="Rudin">Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, ISBN 0-07-100276-6.</ref> ипак није сваки подскуп мерљив.<ref>Gerald Folland, Real Analysis: modern techniques and their applications, John Wiley & Sons, Inc., 1999,Page 20,ISBN 0-471-31716-0</ref> Генерално, површина у вишој математици види се као специјалан случај [[Запремина |запремине]] за дводимензионалне регије.<ref name="MathWorld">{{cite web|url = http://mathworld.wolfram.com/Area.html|title = Area|publisher = [[Wolfram MathWorld]]|author = [[Eric W. Weisstein]]|accessdate = 3 July 2012}}</ref>
Површина може бити дефинисана кроз употребу аксиома, дефинишући је као функцију колекције одређених равних фигура у скуп реалних бројева. Може бити доказано да таква функција постоји.
== Формална дефиниција ==
Појам „површине” су дефинише [[аксиом]]има. Површина може бити дефинисана као функција из колекције -{M}- specijalne vrste равних фигура (названи мерљиви скупови) ка скупу реалних бројева који задовољавају следеће особине:
* За све -{''S''}- у -{''M''}-, -{''a''(''S'') ≥ 0}-.
* Ако су -{''S''}- и -{''T''}- у -{''M''}- тада су и -{''S'' ∪ ''T''}- и -{''S'' ∩ ''T''}-, и такође -{''a''(''S''∪''T'') = ''a''(''S'') + ''a''(''T'') − ''a''(''S''∩''T'')}-.
* Ако су -{''S''}- и -{''T''}- у -{''M''}- са -{''S'' ⊆ ''T''}- тада је -{''T'' − ''S''}- у -{''M''}- и -{''a''(''T''−''S'') = ''a''(''T'') − ''a''(''S'')}-.
* Ако је скуп -{''S''}- у -{''M''}- и -{''S''}- је конгруентно са -{''T''}- тада -{''T''}- је такође у -{''M''}- и -{''a''(''S'') = ''a''(''T'')}-.
* Сваки правоугаоник -{''R''}- је у -{''M''}-. Ако правоугаоник има дужину -{''h''}- и ширину -{''k''}- тада је -{''a''(''R'') = ''hk''}-.
* Нека ''Q'' буде скуп затворен између двије степ регије -{''S''}- и -{''T''}-. Степ регија је формирана од ограничене уније сусједних правоугаоника који се налазе на истој бази, нпр. -{''S'' ⊆ ''Q'' ⊆ ''T''}-. Ако постоји уникатан број -{''c''}- такав да је -{''a''(''S'') ≤ c ≤ ''a''(''T'')}- за све такве степ регије -{''S''}- и -{''T''}-, тада је -{''a''(''Q'') = ''c''}-.
Може бити доказано да таква површинска функција доиста постоји.<ref name="Moise">{{cite book|last = Moise|first = Edwin|title = Elementary Geometry from an Advanced Standpoint|url = http://books.google.com/?id=7nUNAQAAIAAJ|accessdate = 15 July 2012|year = 1963|publisher = Addison-Wesley Pub. Co.|isbn = |page = }}</ref>
== Историја ==
=== Површина круга ===
In the 5th century BCE, [[Hippocrates of Chios]] was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his [[Quadrature (mathematics)|quadrature]] of the [[lune of Hippocrates]],<ref name="heath">{{citation|first=Thomas L.|last=Heath|authorlink=Thomas Little Heath|title=A Manual of Greek Mathematics|publisher=Courier Dover Publications |year=2003|isbn=0-486-43231-9 |pages=121–132|url=https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA121}}</ref> but did not identify the [[constant of proportionality]]. [[Eudoxus of Cnidus]], also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.<ref>{{cite book|url=http://www.stewartcalculus.com/media/8_home.php|title=Single variable calculus early transcendentals.|last=Stewart|first=James|publisher=Brook/Cole|year=2003|isbn=0-534-39330-6|edition=5th.|location=Toronto ON|page=3|quote=However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a circle: <math>A= \pi r^2.</math>}}</ref>
Subsequently, Book I of [[Euclid's Elements|Euclid's ''Elements'']] dealt with equality of areas between two-dimensional figures. The mathematician [[Archimedes]] used the tools of [[Euclidean geometry]] to show that the area inside a circle is equal to that of a [[right triangle]] whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book ''[[Measurement of a Circle]]''. (The circumference is 2{{pi}}''r'', and the area of a triangle is half the base times the height, yielding the area {{pi}}''r''<sup>2</sup> for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) with [[Area of a disk#Archimedes' doubling method|his doubling method]], in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular [[hexagon]], then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with [[circumscribed polygon]]s).
Swiss scientist [[Johann Heinrich Lambert]] in 1761 proved that [[pi|π]], the ratio of a circle's area to its squared radius, is [[irrational number|irrational]], meaning it is not equal to the quotient of any two whole numbers.<ref name=Arndt>{{cite book| last=Arndt |first=Jörg |last2=Haene l|first2=Christoph |title=Pi Unleashed| publisher=Springer-Verlag|year=2006| isbn=978-3-540-66572-4 <!--isbn only volume 1--> |url= https://books.google.com/?id=QwwcmweJCDQC&printsec=frontcover#v=onepage&q&f=false |ref=harv |accessdate=2013-06-05}} English translation by Catriona and David Lischka.</ref> In 1794 French mathematician [[Adrien-Marie Legendre]] proved that π<sup>2</sup> is irrational; this also proves that π is irrational.<ref>{{citation|first=Howard|last=Eves|title=An Introduction to the History of Mathematics|edition=6th|year=1990|publisher=Saunders|isbn=0-03-029558-0|page=121}}</ref> In 1882, German mathematician [[Ferdinand von Lindemann]] proved that π is [[transcendental number|transcendental]] (not the solution of any [[polynomial equation]] with rational coefficients), confirming a conjecture made by both [[Adrien-Marie Legendre|Legendre]] and Euler.<ref name=Arndt/>{{rp|p. 196}}
=== Површина троугла ===
[[Hero of Alexandria|Heron (or Hero) of Alexandria]] found what is known as [[Heron's formula]] for the area of a triangle in terms of its sides, and a proof can be found in his book, ''Metrica'', written around 60 CE. It has been suggested that [[Archimedes]] knew the formula over two centuries earlier,<ref>{{cite book
| author=Heath, Thomas L.
| title=A History of Greek Mathematics (Vol II)
| publisher=Oxford University Press
| year=1921
| pages=321–323}}</ref> and since ''Metrica'' is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.<ref>{{MathWorld |urlname=HeronsFormula |title=Heron's Formula}}</ref>
In 499 [[Aryabhata]], a great [[mathematician]]-[[astronomer]] from the classical age of [[Indian mathematics]] and [[Indian astronomy]], expressed the area of a triangle as one-half the base times the height in the ''[[Aryabhatiya]]'' (section 2.6).
A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in ''Shushu Jiuzhang'' ("[[Mathematical Treatise in Nine Sections]]"), written by [[Qin Jiushao]].
=== Квадрилатерална површина ===
In the 7th century CE, [[Brahmagupta]] developed a formula, now known as [[Brahmagupta's formula]], for the area of a [[cyclic quadrilateral]] (a [[quadrilateral]] [[inscribed figure|inscribed]] in a circle) in terms of its sides. In 1842 the German mathematicians [[Carl Anton Bretschneider]] and [[Karl Georg Christian von Staudt]] independently found a formula, known as [[Bretschneider's formula]], for the area of any quadrilateral.
=== Општа површина полигона ===
The development of [[Cartesian coordinate system|Cartesian coordinates]] by [[René Descartes]] in the 17th century allowed the development of the [[Shoelace formula|surveyor's formula]] for the area of any polygon with known [[vertex (geometry)|vertex]] locations by [[Gauss]] in the 19th century.
=== Површине утврђене коришћењем рачуна ===
The development of [[integral calculus]] in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an [[ellipse#Area|ellipse]] and the [[surface area]]s of various curved three-dimensional objects.
== Формуле за површину ==
=== Формуле полигона ===
{{Main article|Polygon#Area and centroid}}
For a non-self-intersecting ([[simple polygon|simple]]) polygon, the [[Cartesian coordinate system|Cartesian coordinates]] <math>(x_i, y_i)</math> (''i''=0, 1, ..., ''n''-1) of whose ''n'' [[vertex (geometry)|vertices]] are known, the area is given by the [[shoelace formula|surveyor's formula]]:<ref>{{cite web
|url = http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
|title = Calculating The Area And Centroid Of A Polygon
|last = Bourke
|first = Paul
|date = July 1988
|work =
|publisher =
|accessdate = 6 Feb 2013
}}
</ref>
:<math>A = \frac{1}{2} | \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i) |,</math>
where when ''i''=''n''-1, then ''i''+1 is expressed as [[modular arithmetic|modulus]] ''n'' and so refers to 0.
==== Правоугаоници ====
[[File:RectangleLengthWidth.svg|thumb|right|180px|alt=A rectangle with length and width labelled|The area of this rectangle is {{mvar|lw}}.]]
The most basic area formula is the formula for the area of a [[rectangle]]. Given a rectangle with length {{mvar|l}} and width {{mvar|w}}, the formula for the area is:</big><ref name=AF /><ref>{{cite web|url=http://proofwiki.org/wiki/Area_of_Parallelogram/Rectangle|title=Area of Parallelogram/Rectangle|publisher=ProofWiki.org|accessdate=29 May 2016}}</ref>
:{{bigmath|''A'' {{=}} ''lw''}} (rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, as {{math|''l'' {{=}} ''w''}} in the case of a square, the area of a square with side length {{mvar|s}} is given by the formula:<ref name=MathWorld/><ref name=AF/><ref>{{cite web|url=http://proofwiki.org/wiki/Area_of_Square|title=Area of Square|publisher=ProofWiki.org|accessdate=29 May 2016}}</ref>
:{{bigmath|''A'' {{=}} ''s''<sup>2</sup>}} (square).
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a [[definition]] or [[axiom]]. On the other hand, if [[geometry]] is developed before [[arithmetic]], this formula can be used to define [[multiplication]] of [[real number]]s.
[[File:ParallelogramArea.svg|thumb|left|180px|alt=A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle|Equal area figures.]]
==== Дисекција, паралелограми и троуглови ====
{{Main article|Triangle#Computing the area of a triangle|Parallelogram#Area formula}}
Most other simple formulas for area follow from the method of [[dissection (geometry)|dissection]].
This involves cutting a shape into pieces, whose areas must [[addition|sum]] to the area of the original shape.
For an example, any [[parallelogram]] can be subdivided into a [[trapezoid]] and a [[right triangle]], as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:<ref name=AF/>
:{{bigmath|''A'' {{=}} ''bh''}} <big> (parallelogram).</big>
[[File:TriangleArea.svg|thumb|right|180px|alt=A parallelogram split into two equal triangles|Two equal triangles.]]However, the same parallelogram can also be cut along a [[diagonal]] into two [[congruence (geometry)|congruent]] triangles, as shown in the figure to the right. It follows that the area of each [[triangle]] is half the area of the parallelogram:<ref name=AF/>
:<math>A = \frac{1}{2}bh</math> <big> (triangle).</big>
Similar arguments can be used to find area formulas for the [[trapezoid]]<ref>{{citation|title=Problem Solving Through Recreational Mathematics|first1=Bonnie|last1=Averbach|first2=Orin|last2=Chein|publisher=Dover|year=2012|isbn=978-0-486-13174-0|page=306|url=https://books.google.com/books?id=Dz_CAgAAQBAJ&pg=PA306}}</ref> as well as more complicated [[polygon]]s.<ref>{{citation|title=Calculus for Scientists and Engineers: An Analytical Approach|first=K. D.|last=Joshi|publisher=CRC Press|year=2002|isbn=978-0-8493-1319-6|page=43|url=https://books.google.com/books?id=5SDcLHkelq4C&pg=PA43}}</ref>
=== Површина закривљених облика ===
==== Кругови ====
[[File:CircleArea.svg|thumb|right|alt=A circle divided into many sectors can be re-arranged roughly to form a parallelogram|A circle can be divided into [[Circular sector|sectors]] which rearrange to form an approximate [[parallelogram]].]]
{{main article|Area of a circle}}
The formula for the area of a [[circle]] (more properly called the area enclosed by a circle or the area of a [[disk (mathematics)|disk]]) is based on a similar method. Given a circle of radius {{math|''r''}}, it is possible to partition the circle into [[Circular sector|sectors]], as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is {{math|''r''}}, and the width is half the [[circumference]] of the circle, or {{math|π''r''}}. Thus, the total area of the circle is {{math|π''r''<sup>2</sup>}}:<ref name=AF/>
:{{bigmath|''A'' {{=}} π''r''<sup>2</sup>}} <big> (circle).</big>
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The [[limit (mathematics)|limit]] of the areas of the approximate parallelograms is exactly {{math|π''r''<sup>2</sup>}}, which is the area of the circle.<ref name=Surveyor>{{cite journal|last1=Braden|first1=Bart|date=September 1986|title= The Surveyor's Area Formula|journal=The College Mathematics Journal|volume=17|issue=4|pages=326–337|publisher=|doi=10.2307/2686282|url=http://www.maa.org/pubs/Calc_articles/ma063.pdf|accessdate=15 July 2012}}</ref>
This argument is actually a simple application of the ideas of [[calculus]]. In ancient times, the [[method of exhaustion]] was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to [[integral calculus]]. Using modern methods, the area of a circle can be computed using a [[definite integral]]:
:<math>A \;=\;2\int_{-r}^r \sqrt{r^2 - x^2}\,dx \;=\; \pi r^2.</math>
==== Елипсе ====
{{main article|Ellipse#Area}}
The formula for the area enclosed by an [[ellipse]] is related to the formula of a circle; for an ellipse with [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes {{math|''x''}} and {{math|''y''}} the formula is:<ref name=AF /><ref>{{cite web|url=http://proofwiki.org/wiki/Area_of_Parallelogram/Rectangle|title=Area of Parallelogram/Rectangle|publisher=ProofWiki.org|accessdate=29 May 2016}}</ref>
:<math>A = \pi xy .</math>
==== Површина области ====
{{main article|Surface area}}
[[File:Archimedes sphere and cylinder.svg|right|thumb|180px|alt=A blue sphere inside a cylinder of the same height and radius|[[Archimedes]] showed that the surface area of a [[sphere]] is exactly four times the area of a flat [[disk (mathematics)|disk]] of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a [[cylinder (geometry)|cylinder]] of the same height and radius.]]
Most basic formulas for [[surface area]] can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a [[cylinder (geometry)|cylinder]] (or any [[prism (geometry)|prism]]) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a [[cone (geometry)|cone]], the side surface can be flattened out into a sector of a circle, and the resulting area computed.
The formula for the surface area of a [[sphere]] is more difficult to derive: because a sphere has nonzero [[Gaussian curvature]], it cannot be flattened out. The formula for the surface area of a sphere was first obtained by [[Archimedes]] in his work ''[[On the Sphere and Cylinder]]''. The formula is:<ref name=MathWorldSurfaceArea/>
:{{bigmath|''A'' {{=}} 4''πr''<sup>2</sup>}} <big> (sphere),</big>
where {{math|''r''}} is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to [[calculus]].
=== Опште формуле ===
==== Површине дводимензионалних фигура ====
* A [[triangle]]: <math>\tfrac12Bh</math> (where ''B'' is any side, and ''h'' is the distance from the line on which ''B'' lies to the other vertex of the triangle). This formula can be used if the height ''h'' is known. If the lengths of the three sides are known then ''[[Heron's formula]]'' can be used: <math>\sqrt{s(s-a)(s-b)(s-c)}</math> where ''a'', ''b'', ''c'' are the sides of the triangle, and <math>s = \tfrac12(a + b + c)</math> is half of its perimeter.<ref name=AF/> If an angle and its two included sides are given, the area is <math>\tfrac12 a b \sin(C)</math> where {{math|''C''}} is the given angle and {{math|''a''}} and {{math|''b''}} are its included sides.<ref name=AF/> If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of <math>\tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)</math>. This formula is also known as the [[shoelace formula]] and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points ''(x<sub>1</sub>,y<sub>1</sub>)'', ''(x<sub>2</sub>,y<sub>2</sub>)'', and ''(x<sub>3</sub>,y<sub>3</sub>)''. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use [[calculus]] to find the area.
* A [[simple polygon]] constructed on a grid of equal-distanced points (i.e., points with [[integer]] coordinates) such that all the polygon's vertices are grid points: <math>i + \frac{b}{2} - 1</math>, where ''i'' is the number of grid points inside the polygon and ''b'' is the number of boundary points.<ref name=Pick>{{cite journal |last=Trainin |first=J. |title=An elementary proof of Pick's theorem |journal=[[Mathematical Gazette]] |volume=91 |issue=522 |date=November 2007 |pages=536–540}}</ref> This result is known as [[Pick's theorem]].<ref name=Pick/>
==== Површина у рачуну ====
[[File:Integral as region under curve.svg|left|thumb|280px|alt=A diagram showing the area between a given curve and the x-axis|Integration can be thought of as measuring the area under a curve, defined by ''f''(''x''), between two points (here ''a'' and ''b'').]]
[[File:Areabetweentwographs.svg|thumb|287px|alt=A diagram showing the area between two functions|The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions]]
* The area between a positive-valued curve and the horizontal axis, measured between two values ''a'' and ''b'' (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from ''a'' to ''b'' of the function that represents the curve:<ref name=MathWorld/>
:<math> A = \int_a^{b} f(x) \, dx.</math>
* The area between the [[graph of a function|graphs]] of two functions is [[equality (mathematics)|equal]] to the [[integral]] of one [[function (mathematics)|function]], ''f''(''x''), [[subtraction|minus]] the integral of the other function, ''g''(''x''):
:<math> A = \int_a^{b} ( f(x) - g(x) ) \, dx, </math> where <math> f(x) </math> is the curve with the greater y-value.
* An area bounded by a function ''r'' = ''r''(θ) expressed in [[polar coordinates]] is:<ref name=MathWorld/>
:<math>A = {1 \over 2} \int r^2 \, d\theta. </math>
* The area enclosed by a [[parametric curve]] <math>\vec u(t) = (x(t), y(t)) </math> with endpoints <math> \vec u(t_0) = \vec u(t_1) </math> is given by the [[line integral]]s:
::<math> \oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt </math>
(see [[Green's theorem]]) or the ''z''-component of
:<math>{1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.</math>
==== Ограничена површина између две квадратне функције ====
To find the bounded area between two [[quadratic function]]s, we subtract one from the other to write the difference as
:<math>f(x)-g(x)=ax^2+bx+c=a(x-\alpha)(x-\beta)</math>
where ''f''(''x'') is the quadratic upper bound and ''g''(''x'') is the quadratic lower bound. Define the [[discriminant]] of ''f''(''x'')-''g''(''x'') as
:<math>\Delta=b^2-4ac.</math>
By simplifying the integral formula between the graphs of two functions (as given in the section above) and using [[Vieta's formulas|Vieta's formula]], we can obtain<ref>{{cite book|title=Matematika|url=https://books.google.com/books?id=NFkVfrZBqpUC&pg=PA51|publisher=PT Grafindo Media Pratama|isbn=978-979-758-477-1|pages=51–}}</ref><ref>{{cite book|title=Get Success UN +SPMB Matematika|url=https://books.google.com/books?id=uwqvITs8OaUC&pg=PA157|publisher=PT Grafindo Media Pratama|isbn=978-602-00-0090-9|pages=157–}}</ref>
:<math>A=\frac{\Delta\sqrt{\Delta}}{6a^2}=\frac{a}{6}(\beta-\alpha)^3,\qquad a\neq0.</math>
The above remains valid if one of the bounding functions is linear instead of quadratic.
==== Површина тродимензионалних фигура ====
* [[Cone]]:<ref name=MathWorldCone>{{cite web|url=http://mathworld.wolfram.com/Cone.html|title=Cone|publisher=[[Wolfram MathWorld]]|authorlink=Eric W. Weisstein|author=Weisstein, Eric W.|accessdate=6 July 2012}}</ref> <math>\pi r\left(r + \sqrt{r^2 + h^2}\right)</math>, where ''r'' is the radius of the circular base, and ''h'' is the height. That can also be rewritten as <math>\pi r^2 + \pi r l </math><ref name=MathWorldCone/> or <math>\pi r (r + l) \,\!</math> where ''r'' is the radius and ''l'' is the slant height of the cone. <math>\pi r^2 </math> is the base area while <math>\pi r l </math> is the lateral surface area of the cone.<ref name=MathWorldCone/>
* [[cube]]: <math>6s^2</math>, where ''s'' is the length of an edge.<ref name=MathWorldSurfaceArea/>
* [[cylinder]]: <math>2\pi r(r + h)</math>, where ''r'' is the radius of a base and ''h'' is the height. The ''2<math>\pi</math>r'' can also be rewritten as ''<math>\pi</math> d'', where ''d'' is the diameter.
* [[Prism (geometry)|prism]]: 2B + Ph, where ''B'' is the area of a base, ''P'' is the perimeter of a base, and ''h'' is the height of the prism.
* [[pyramid (geometry)|pyramid]]: <math>B + \frac{PL}{2}</math>, where ''B'' is the area of the base, ''P'' is the perimeter of the base, and ''L'' is the length of the slant.
* [[rectangular prism]]: <math>2 (\ell w + \ell h + w h)</math>, where <math>\ell</math> is the length, ''w'' is the width, and ''h'' is the height.
==== Општа формула за површину ====
The general formula for the surface area of the graph of a continuously differentiable function <math>z=f(x,y),</math> where <math>(x,y)\in D\subset\mathbb{R}^2</math> and <math>D</math> is a region in the xy-plane with the smooth boundary:
: <math> A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy. </math>
An even more general formula for the area of the graph of a [[parametric surface]] in the vector form <math>\mathbf{r}=\mathbf{r}(u,v),</math> where <math>\mathbf{r}</math> is a continuously differentiable vector function of <math>(u,v)\in D\subset\mathbb{R}^2</math> is:<ref name="doCarmo"/>
: <math> A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. </math>
=== Основне формуле ===
Линија 61 ⟶ 230:
|<math>r</math> и <math>\theta</math> су полупречник и угао (у [[радијан]]има).
|}
== Мерне јединице ==
Линија 70 ⟶ 237:
* -{1 cm<sup>2</sup>}- = 0.0001 -{m}-<sup>2</sup> = 10<sup>-4</sup>-{m}-<sup>2</sup> (ретко се користи)
* -{1 mm<sup>2</sup>}- = 0.000001 -{m}-<sup>2</sup> = 10<sup>-6</sup>-{m}-<sup>2</sup> (користи се за мерење површине пресека жице у [[Електротехника|електротехници]])
За мерење површине терена користе се веће мере:
Линија 79 ⟶ 245:
== Види још ==
* [[Површ]]
== Референце ==
{{reflist|30em}}
== Спољашње везе ==
{{Commonscat|Area}}
* -{[http://blog.thinkwell.com/2010/07/6th-grade-math-surface-area.html Surface Area Video] at Thinkwell}-
{{Authority control}}
[[Категорија:Физичке величине]]
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