ÄéðëÜ Ïëïêëçñþìáôá

ÄéðëÜ ïëïêëçñþìáôá - ÂáóéêÝò Éäéüôçôåò

Ïñéóìüò.

¸óôù üôé ç f[x,y] åßíáé ìßá óõíå÷Þò óõíÜñôçóç ïñéóìÝíç óå Ýíá öñáãìÝíï äéäéÜóôáôï ÷ùñßï R⊂(ℝ^2). Áò èåùñÞóïõìå üôé ôï R êáëýðôåôáé áðü Ýíá ïñèïãþíéï ðëÝãìá ôï ïðïßï áðáñôßæåôáé áðü åõèåßåò ðáñÜëëçëåò óôïí x-Üîïíá êáé  y-Üîïíá. ÁõôÝò ïé åõèåßåò ÷ùñßæïõí ôï ÷ùñßï R óå ìéêñÜ ôåôñÜãùíá ÷ùñßá Δ(Subscript[A, k])=Δ(Subscript[x, k]) · Δ(Subscript[y, k]).  Èåùñïýìå åêåßíá ôá (Subscript[ΔA, k]) ôá ïðïßá ðåñéÝ÷ïíôáé ïëüêëçñá óôï ÷ùñßï R.  ÅðéëÝãïíôáò Ýíá ôõ÷áßï óçìåßï {(Subscript[x, k]), (Subscript[y, k])} óå êÜèå Δ(Subscript[A, k]), ó÷çìáôßæïõìå ôï Üèñïéóìá    (Underoverscript[∑ , (k = 1), n]) f [(Subscript[x, k,]) (Subscript[y, k])]  Δ(Subscript[A, k]) .  (Áõôü ðñÝðåé íá óáò èõìßæåé ôï Üèñïéóìá Riemann ðïõ åßäáôå óôïí Áðåéñïóôéêü Ëïãéóìü ÉÉ)

Ôüôå ïñßæåôáé ôï äéðëü ïëïêëÞñùìá ôçò óõíÜñôçóçò  f[x,y]  åðß ôïõ ÷ùñßïõ R íá åßíáé

∫(∫_(R )) f[x, y]dA =  (lim_(ΔA ->  0)) (∑_(k = 1)^n) 
                f [(Subscript[x, k,]) (Subscript[y, k])]  Δ(Subscript[A, k])   
(üðïõ ΔA åßíáé ôï ìåãáëýôåñï áðü ôá (Subscript[ΔA, k]))

Ç óõíÝ÷åéá ôçò f[x,y] åßíáé éêáíÞ óõíèÞêç ãéá ôçí ýðáñîç ôïõ äéðëïý ïëïêëçñþìáôïò áëëÜ ü÷é áíáãêáßá. Ôï ðáñáðÜíù üñéï õðÜñ÷åé êáé ãéá ðïëëÝò ìç óõíå÷åßò óõíáñôÞóåéò.

Éäéüôçôåò äéðëïý ïëïêëçñþìáôïò

¼ðùò êáé óôá ïëïêëçñþìáôá ìéÜò ìåôáâëçôÞò ôá äéðëÜ ïëïêëçñþìáôá ôùí óõíå÷þí óõíáñôÞóåùí Ý÷ïõí áñêåôÝò áëãåâñéêÝò éäéüôçôåò ðïõ åßíáé ÷ñÞóéìåò óôïõò õðïëïãéóìïýò êáé óôéò åöáñìïãÝò. Ïé óçìáíôéêüôåñåò åßíáé ïé åîÞò:

1.   ∫(∫_(R ))k f[x, y]dA = k ∫(∫_(R ))  f[x, y] dA    
    üðïõ k óôáèåñüò ðñáãìáôéêüò  

2.  ∫(∫_(R ))(  f[x, y] + g[x, y] )dA = 

                ∫(∫_(R )) f[x, y] dA + ∫(∫_(R )) g[x, y]  dA

3. ∫(∫_(R ))(  f[x, y] - g[x, y] )dA = 

                 ∫(∫_(R )) f[x, y] dA - ∫(∫_(R )) g[x, y]  dA

4. ∫(∫_(R )) f[x, y] dA ⩾ 0,  Üí  f [x,y] ⩾ 0  åðß ôïõ R

5. ∫(∫_(R )) f[x, y] dA ⩾∫(∫_(R )) g[x, y]dA, 
  
   áí   f [x,y] ⩾ g[x,y] åðß ôïõ R

ÅðéðëÝïí ôá äéðëÜ ïëïêëçñþìáôá óõìðåñéöÝñïíôáé "ðñïóèåôéêÜ" üôáí ôï ÷ùñßï R åßíáé ç Ýíùóç äýï ìç åðéêáëõðôüìåíùí ÷ùñßùí (Subscript[R, 1]) êáé  (Subscript[R, 2]).

6. ∫ (∫_(R )) f[x, y] dA  =  ∫ (∫_((Subscript[R, 1]) )) f[x, y] dA  + ∫ (∫_((Subscript[(R ), 2]))) f[x, y] dA

Ôï äéðëü ïëïêëÞñùìá ùò üãêïò óôåñåïý

¼ôáí ç óõíÜñôçóç f[x,y] åßíáé ìç áñíçôéêÞ åðß åíüò ÷ùñßïõ R, ôüôå ôï äéðëü ïëïêëÞñùìá ôçò f[x,y] åðß ôïõ R åêöñÜæåé ôïí üãêï ôïõ óôåñåïý ðïõ öñÜóóåôáé ðñïò ôá êÜôù áðü ôï ÷ùñßï  R êáé ðñïò ôá ðÜíù áðü ôçí åðéöÜíåéá z = f[x,y] (üðïõ ïé ëÝîåéò ðÜíù êáé êÜôù ÷ñçóéìïðïéïýíôáé áíáöïñéêÜ ìå ôï ìÝãåèïò ôçò óõíôåôáãìÝíçò z). Ìå Üëëá ëüãéá

¼ãêïò êÜôù áðü ôï ãñÜöçìá ôçò f[x,y] = 
                           ∫(∫_(R )) f[x , y] dA,  áí f[x,y] ≥ 0 åðß ôïõ R.

Èåþñçìá Fubini ãéá ôá äéðëÜ ïëïêëçñþìáôá

Ôï èåþñçìá Fubini ìáò ëÝåé üôé ôï äéðëü ïëïêëÞñùìá ìðïñåß íá õðïëïãéóèåß ìå äéáäï÷éêÝò ïëïêëçñþóåéò ùò ðñïò ôéò äýï ìåôáâëçôÝò x êáé y êáé ìÜëéóôá áíåîáñôÞôùò óåéñÜò, õðü ôçí ðñïûðüèåóç üôé ç óõíÜñôçóç åßíáé óõíå÷Þò.

Èåþñçìá Fubini
¸óôù f[x, y] óõíå÷Þò óõíÜñôçóç åðß ôïõ ÷ùñßïõ R.

1. ÅÜí ôï ÷ùñßï R êáèïñßæåôáé áðü ôéò áíéóüôçôåò a ⩽ x ⩽ b,  á[x] ⩽ y ⩽ â[x] , ìå  á[x]  êáé â[x] óõíå÷åßò åðß ôïõ [a , b], ôüôå

             ∫(∫_(R )) f[x, y] dA = (∫_a^b)(∫_(á[x] )^(â[x]))f[x, y] dy dx
                                                  
                                                   
üðïõ  (∫_a^b)(∫_(á[x] )^(â[x]))f[x, y] dy dx   åßíáé äéáäï÷éêü ïëïêëÞñùìá êáé ôï ðñþôï ïëïêëÞñùìá ðñïêýðôåé ïëïêëçñþíïíôáò ðÜíù áðü ôïìÝò ôïõ R ðáñÜëëçëåò ìå ôïí y-Üîïíá (óõãêåêñéììÝíá ôï åõèýãñáììï ôìÞìá  á[x] ≤ y ≤ â[x]).

2. ÅÜí ôï ÷ùñßï R êáèïñßæåôáé áðü ôéò áíéóüôçôåò c ⩽ y ⩽ d,  ã[y] ⩽ x ⩽ ä[y] , ìå  ã[y]  êáé  ä[y]  óõíå÷åßò åðß ôïõ [c , d], ôüôå 

            ∫(∫_(R )) f[x, y]  dA = (∫_c^d)(∫_(ã[y] )^(ä[y] ))f[x, y] dx dy
                                               
üðïõ (∫_c^d)(∫_(ã[y] )^(ä[y] ))f[x, y] dx dy   åßíáé äéáäï÷éêü ïëïêëÞñùìá êáé ôï ðñþôï ïëïêëÞñùìá ðñïêýðôåé ïëïêëçñþíïíôáò ðÜíù áðü ôïìÝò ôïõ R ðáñÜëëçëåò ìå ôïí x-Üîïíá (óõãêåêñéììÝíá óôï åõèýãñáììï ôìÞìá  ã[y] ≤ x ≤ ä[y]).

ÏëïêëÞñùóç åðß ïñèïãùíßïõ Ðáñáë/ãñÜììïõ

ÐáñÜäåéãìá 1  (¼ãêïò óôåñåïý ðïõ öñÜóóåôáé áðü ìßá åðéöÜíåéá: ÏëïêëÞñùóç åðß ïñèïãùíßïõ ðáñáë/ìïõ)

Èá õðïëïãßóïõìå ôïí üãêï ôïõ óôåñåïý ðïõ ïñéïèåôåßôáé ðñïò ôá ðÜíù (üðùò ðñïçãïõìÝíùò, ïé ëÝîåéò ðÜíù êáé êÜôù ÷ñçóéìïðïéïýíôáé áíáöïñéêÜ ìå ôï ìÝãåèïò ôçò óõíôåôáãìÝíçò z) áðü ôçí åðéöÜíåéá  f[x,y] = (x^2)+ (y^(2 )) êáé ðñïò ôá êÜôù áðü ôï ïñèïãþíéï [0,2] x [0,1] (óôï xy-åðßðåäï).

Ç åéêüíá ôçò åðéöÜíåéáò ìáæß ìå ôï ïñèïãþíéï åßíáé ç åîÞò:

Clear[x, y, surface, f]
f[x_, y_] =  (x^2) +  (y^2);
\(\(surface = 
        ParametricPlot3D[
{{x, y, \ f[x, y]}, {x, y, 0}}, {x, 0, 2}, {y, 0, 
            1}, 
ViewPoint -> {2.5\ , 1.7, \ 1.4}, 
BoxRatios -> {2, 1, 2}, 
          AxesLabel -> {"x", "y", "z"}];\)\)

Ìðïñïýìå íá âñïýìå ôïí üãêï ôïõ óôåñåïý êÜôù áðü ôçí åðéöÜíåéá êáé ðÜíù áðü ôï ïñèïãþíéï "êüâïíôáò" ôï óôåñåü óå öÝôåò. Ðéü óõãêåêñéììÝíá, èåùñïýìå åãêÜñóéåò ôïìÝò êÜôù áðü ôçí åðéöÜíåéá üðùò öáßíåôáé óôï åðüìåíï ó÷Þìá.

Clear[y,gr1,Q];
Q[s_,x_]={x,y,0} + s ({x,y,f[x,y]}- {x,y,0});
y=.4;

gr1=ParametricPlot3D[Q[s,x],{s,0,1},{x,0,2},
BoxRatios->{2,1,2},PlotRange->All,
DisplayFunction->Identity];


Show[surface,gr1,ViewPoint->{2.5 ,1.7, 1.4},
PlotRange->All,
BoxRatios->{2,1,2},AxesLabel->{"x","y","z"}];

ÎÝñïõìå Þäç ðùò íá âñïýìå ôï åìâáäüí áõôÞò ôçò åãêÜñóéáò ôïìÞò ìå ÷ñÞóç ïñéóìÝíïõ ïëïêëçñþìáôïò (ìéáò ìåôáâëçôÞò).   Ôï åìâáäüí ôçò åãêÜñóéáò ôïìÞò (ãéá y=(2/5) = 0.40) åßíáé:

y =  .4;
(∫_0^2)f[x, y] \[DifferentialD]x

Ôï åìâáäüí ìéáò ôõðéêÞò åãêÜñóéáò ôïìÞò ãéá ôõ÷áßá ôéìÞ ôçò ìåôáâëçôÞò y ìåôáîý 0 êáé 1 åßíáé:

Clear[y, g]
g[y_]= (∫_0^2)f[x, y] \[DifferentialD]x

ÐáñáêÜôù ó÷åäéÜæïíôáé åãêÜñóéåò ôïìÝò ãéá äéÜöïñåò ôéìÝò ôçò ìåôáâëçôÞò y ìåôáîý 0 êáé 1.

Clear[y,gr1,Q];
Q[s_,x_]={x,y,0} + s ({x,y,f[x,y]}- {x,y,0});
y=-.35;
While[ y<=.7,y=y+.35;
Clear[gr1];
gr1=ParametricPlot3D[Q[s,x],{s,0,1},{x,0,2},
BoxRatios->{2,1,2},PlotRange->All,
DisplayFunction->Identity];


Show[surface,gr1,ViewPoint->{2.5 ,1.7, 1.4},
PlotRange->All,
BoxRatios->{2,1,2},AxesLabel->{"x","y","z"}]];

Åßíáé öáíåñü üôé ïé åãêÜñóéåò ôïìÝò "óáñþíïõí" ôï óôåñåü êÜôù áðü ôçí åðéöÜíåéá. Ùò åê ôïýôïõ, ãéá íá âñïýìå ôïí üãêï  ôïõ óôåñåïý  (ðïõ óáñþíïõí ïé åãêÜñóéåò ôïìÝò), ïëïêëçñþíïõìå ôï åìâáäüí ìéÜò ôõðéêÞò åãêÜñóéáò ôïìÞò ðïõ âñÞêáìå ðáñáðÜíù, äçëáäÞ ôï g[y] (8/3) + 2 (y^2) áðü ôï y=0 Ýùò ôï y=1.

Clear[y]
(∫_0^1)g[y] \[DifferentialD]y

Åßäáìå ëïéðüí ðùò ôï äéðëü ïëïêëÞñùìá õðïëïãßæåôáé ìå äýï áëëåðÜëëçëá ïëïêëçñþìáôá. Óôï Mathematica  áõôü ãßíåôáé ìå ôçí åîÞò åíôïëÞ:

(∫_0^1)(∫_0^2)f[x, y] \[DifferentialD]x\[DifferentialD]y
Integrate[
      f[x,  y],  {y,  0,  1},  {x,  0,  2}]

ÐñïóÝîôå üôé óôïí êþäéêá (Integrate) ï õðïëïãéóìüò ãßíåôáé ðñþôá ùò ðñïò ôçí ìåôáâëçôÞ ðïõ áíáöÝñåôáé ôåëåõôáßá (åí áíôéèÝóåé ìå ôïí êëáóóéêü óõìâïëéóìü \(∫\(∫ dx dy\)\)).

ÏëïêëÞñùóç åðß öñáãìÝíùí ÷ùñßùí ðïõ äåí åßíáé ïñèãþíéá.

ÐáñÜäåéãìá 2   (¼ãêïò óôåñåïý ðïõ öñÜóóåôáé áðü åðéöÜíåéá ìå ÷ñÞóç êáôáêüñõöùí ôïìþí--ïëïêëÞñùóç ðñþôá ùò ðñïò y)

Ï ðñáêÜôù êþäéêáò ó÷åäéÜæåé ôçí åðéöÜíåéá f[x, y] =  ((x )^4) (y^2) + 2 êáé ôïí äßóêï (x^2) + (y^2) ≤ 1. Ç ãñáöéêÞ ðáñÜóôáóç ãßíåôáé ìå ôçí åíôïëÞ ParametricPlot3D.

Clear[x, y, t, r]
x =  r Cos[t];
y =  r Sin[t];
f[x_, y_] =  ((x )^4) (y^2) + 2;
\(\(ParametricPlot3D[
{{\ x, \ y, \ f[x, y]}, {x, \ \ y, 0}}, {t, 0, 
          2  π}, {r, 0, 1}];\)
\)

Ãéá íá âñïýìå ôïí üãêï ôïõ óôåñåïý ðïõ öñÜóóåôáé áðü ôïí äßóêï (ðñïò ôá êÜôù) êáé áðü ôï ãñÜöçìá ôçò f[x,y] (ðñïò ôá åðÜíù), èá ÷ñçóéìïðïéÞóïõìå ôï èåþñçìá Fubini êáé åãêÜñóéåò ôïìÝò ðïõ èá Ý÷ïõí âÜóç óôïí äßóêï.

Clear[f, first, x, y, á, â]
f[x_, y_] =  ((x )^4) (y^2) + 2;

â[x_] = (Sqrt[1 -  (x^2)]);
á[
          x_] =  -(Sqrt[1 -  (x^2)]);
(∫_(-1)^1)(∫_(á[x])^(â[x]))f[x, y] \[DifferentialD]y\[DifferentialD]x

Integrate[
      f[x,  y],  {x,  -1,  
        1},  {y,  á[x],  â[x]}]

ÐáñÜäåéãìá 3  (Åìâáäüí ÷ùñßïõ ìå ÷ñÞóç äéðëïý ïëïêëçñþìáôïò)

Èåùñïýìå ôçí åéäéêÞ ðåñßðôùóç êáôÜ ôçí ïðïßáí ç óõíÜñôçóç ôçò åðéöÜíåéáò åßíáé f[x,y]=1 êáé ôï ÷ùñßï óôï xy-åðßðåäï åßíáé ï äßóêïò ðïõ öñÜóóåôáé áðü ôïí êýêëï (x^2) + (y^2) = 9.

Clear[x,y,t,r]
x= r Cos[t];
y= r Sin[t];
f[x_,y_]= 1;
ParametricPlot3D[{{ x, y, f[x,y]},{x,  y,0}},
{t,0,2π},{r,0,3}];

Õðïëïãßæïõìå ôïí üãêï ôïõ óôåñåïý ðïõ öñÜóóåôáé áðü ôïõò äýï äßóêïõò (ôï ïðïßï åßíáé Ýíáò êýëéíäñïò).

Clear[f, first, x, y, á, â]
\(\(f[x_, y_] = 1;\)
\)
â[x_] = (Sqrt[9 - (x^2)])
á[x_] =  -(Sqrt[9 - (x^2)])
Integrate[1, {x, -3, 3}, {y, á[x], â[x]}]

Ôï áðïôÝëåóìá ðïõ âñÞêáìå åßíáé ôï åìâáäüí ôïõ äßóêïõ âÜóçò. Ãéáôß?

ÐáñÜäåéãìá 4 (¼ãêïò óôåñåïý ðïõ öñÜóóåôáé áðü åðéöÜíåéá ìå ÷ñÞóç ïñéæüíôéùí ôïìþí--ïëïêëÞñùóç ðñþôá ùò ðñïò x)

¸óôù R ôï ÷ùñßï ôïõ ðáñáêÜôù ó÷Þìáôïò óôï xy-åðßðåäï.

Clear[x, y];
\(\(ParametricPlot[{y\^2, y}, {y, \(-2\), 2}, 

        Epilog -> Line[{{4, 2}, {4, \(-2\)}}]];\)\)

Õðïëïãßæïõìå ôï äéðëü ïëïêëÞñùìá ôçò óõíÜñôçóçò f[x,y]= Sin[x] Cos[y]  åðß ôïõ R ïëïêëçñþíïíôáò ðñþôá ùò ðñïò x , ìå ôá üñéá ïëïêëÞñùóçò íá åßíáé óõíáñôÞóåéò ôïõ y.

Clear[x, y, t, á, â, first]
\(\(f[x_, y_] = \ Sin[x]\ Cos[y];\)
\)
á[y_] =  (y^2);
â[y_] = 4;
NIntegrate[
      f[x,  y],  {y,  -2,  
        2},  {x,  á[y],  â[y]}]

ÐáñÜäåéãìá 5 (¼ãêïò óôåñåïý ðïõ öñÜóóåôáé áðü äýï åðéöÜíåéåò)

Ìå ôïí ðáñáêÜôù êþäéêá öôéÜ÷íïõìå ôéò ãñáöéêÝò ðáñáóôÜóåéò ôùí óõíáñôÞóåùí  f[x,y] = (1/4) ((x^2) + (y^2)) êáé g[x,y]= 5  - (x^2) - (y^2) ìåãáëþíïíôáò ôï ÷ùñßï óôï xy-åðßðåäï ìÝ÷ñé ïé äýï ãñáöéêÝò ðáñáóôÜóåéò íá ôìçèïýí. Óôçí ïõóßá ëýíïõìå ôçí åîßóùóç
(1/4) ((x^2) + (y^2))= 5  - (x^2) - (y^2) 
(âñßóêïõìå üôé ç éóüôçôá éó÷ýåé ãéá üëá ôá óçìåßá ôïõ êýêëïõ (x^2) + (y^2) = 4).

\(\ \(Clear[x, y, t, r, f, g, a];\)
\)
  f[x_, y_] =  (1/4) ((x^2) + (y^2));
\(\ \ \(g[x_, y_]\  = \ 5\  - x\^2 - y\^2;\)
\ \ 
\ \ \)
  x =  r Cos[t];
\(\ \ \ \(y = \ r\ Sin[t];\)
\ \ \ \)
\(\ \ \ For[a = 1.8, a\  <= 2, \ 
      a = a +  .1, 
\ \ \ ParametricPlot3D[
\ \ \ {{\ x, \ y, \ 
            f[x, y]}, {x, \ \ y, g[x, y]}, {x, y, 0}}, {t, 0, 
          2  Pi}, 
\ \ \ {r, 0, 
          a}, 
\ \ \ PlotRange -> {{\(-2\), 2}, {\(-2\), 2}, {0, 5}}, 
        ViewPoint -> {3.164, 0.500, \(-3.031\)}]]
\ \ \ \)

Ôï ðáñáêÜôù ïëïêëÞñùìá õðïëïãßæåé ôïí üãêï ôïõ óôåñåïý ðïõ öñÜóóåôáé áðü ôá ãñáöÞìáôá ôùí g[x,y] êáé f[x,y].

Clear[x, y]
(∫_(-2)^2)(∫_(-(Sqrt[4 - (x^2)]))^((Sqrt[4 - (x^2)])))(g[x, y] - f[x, y]) \[DifferentialD]y \[DifferentialD]x

ÌÝóç ôéìÞ, ÊÝíôñï, êáé ÊÝíôñï ìÜæáò

ÌÝóç ôéìÞ óõíÜñôçóçò  f[x,y] åðß ÷ùñßïõ R

Ç ìÝóç ôéìÞ ìéÜò óõíÜñôçóçò  f[x,y] åðß ÷ùñßïõ R ïñßæåôáé ùò 
(1/(Area R))∫(∫_(R    ))f[x, y]dxdy

ÐáñÜäåéãìá 6.

Âñßóêïõìå ôçí ìÝóç ôéìÞ ôçò óõíÜñôçóçò f[x,y]=Sin [x]  (y^(2 ))+2  åðß ôïõ ïñèïãùíßïõ [0, π] x [0,π] .

Clear[f, x, y, Area]
f[x_, y_] =  Sin[x] (y^2) + 2 
Area=(∫_0^π)(∫_0^π)1 \[DifferentialD]x\[DifferentialD]y
(1/Area)(∫_0^π)(∫_0^π) 
                f[x, y] \[DifferentialD]x \[DifferentialD]y//N

Clear[f, x, y, geom]
f[x_, y_] =  Sin[x] (y^2) + 2 
geom=(∫_0^π)(∫_0^π)((1/Area) (∫_0^π)(∫_0^π) 
                      f[x, y] \[DifferentialD]x \[DifferentialD]y) \[DifferentialD]x \[DifferentialD]y
(∫_0^π)(∫_0^π) 
          f[x, y] \[DifferentialD]x \[DifferentialD]y

Ç ìÝóç ôéìÞ ôçò óõíÜñôçóçò f[x,y] åðß ôïõ ÷ùñßïõ [0, π] x [0,π]  åßíáé 4.0944 . 

Ç ÃåùìåôñéêÞ åñìçíåßá ôçò ìÝóçò ôéìÞò åßíáé ç åîÞò: áí f åßíáé èåôéêÞ óõíÜñôçóç åðß åíüò ÷ùñßïõ R ôüôå ç ìÝóç ôéìÞ ôçò f óôï R åßíáé ï áñéèìüò åêåßíïò ôïõ ïðïßïõ ôï ãéíüìåíï ìå ôï åìâáäüí ôïõ R éóïýôáé ìå ôï ïëïêëÞñùìá ôçò f åðß ôïõ R.

Ç áíáëõôéêÞ åñìçíåßá ôçò Ýííïéáò ôçò ìÝóçò ôéìÞò åßíáé áíÜëïãç ôùò ãíùóôÞò ìÝóçò ôéìÞò ðïõ ãíùñßæïõìå áðü ôçí áñéèìçôéêÞ üðùò åðéäåéêíýïõí ïé ðáñáêÜôù õðïëïãéóìïß ôïõ ìÝóïõ üñïõ ôçò ôéìÞò  ôçò f[x,y] óå ôõ÷áßá, ðåðåñáóìÝíá ôï ðëÞèïò, óçìåßá.

Èåùñïýìå ðñþôá 50 óçìåßá.

Clear[n,points,values]
n=50;
points= Table[Random[Real,{0.,N[Pi]}],{n},{2}];
values=Apply[f,points,1];
Sum[values[[k]],{k,1,n}]/n

200 óçìåßá.

Clear[n,points,values]
n=200;
points= Table[Random[Real,{0.,N[Pi]}],{n},{2}];
values=Apply[f,points,1];
Sum[values[[k]],{k,1,n}]/n

1000 óçìåßá.

Clear[n,points,values]
n=1000;
points= Table[Random[Real,{0.,N[Pi]}],{n},{2}];
values=Apply[f,points,1];
Sum[values[[k]],{k,1,n}]/n

Åêôåëþíôáò ôïí ßäéï êþäéêá áõîÜíïíôáò ôïí áñéèìü ôùí óçìåßùí èá âñßóêïõìå êÜèå öïñÜ ìéá äéáöïñåôéêÞ ôéìÞ êïíôÜ óôïí áñéèìü  4.0944.

ÊÝíôñï ÷ùñßïõ

Ôï êÝíôñï åíüò R óôï xy-åðßðåäï åßíáé ôï óçìåßï {x*,y*} üðïõ
       x* åßíáé ç ìÝóç ôéìÞ ôçò óõíÜñôçóçò f[x,y] = x åðß ôïõ R   êáé
       y* åßíáé ç ìÝóç ôéìÞ ôçò óõíÜñôçóçò g[x,y] = y åðß ôïõ R.

ÐáñÜäåéãìá 7.

Áò ðÜñïõìå ôçí ðáñáêÜôù ðåñéï÷Þ R óôï xy-åðßðåäï:

Rplot  = Plot[-x^2 + 4, {x, -2, 2}];

Ï ðáñáêÜôù êþäéêáò õðïëïãßæåé ôï êÝíôñï ôïõ R:

Clear[xstar,  ystar, area]
\(area = Integrate[1, {x, \(-2\), 2}, {y, 0, 4 - x^2}]
\)
\(xstar = \(\(Integrate[x, {x, \(-2\), 2}, {y, 0, 4 - x^2}]/
        area\)\(
\)\)\)
\(ystar = \(\(Integrate[y, {x, \(-2\), 2}, {y, 0, 4 - x^2}]/
        area\)\(
\)\)\)

Óçìåéþíïõìå ôï êÝíôñï ìÝóá óôï ÷ùñßï R:

Clear[centroid]
centroid=Graphics[{Blue,PointSize[.025],Point[{xstar,ystar}]}];
Show[Rplot,centroid];

Ç x-óõíôåôáãìÝíç ôïõ êÝíôñïõ åßíáé ßóç ìå ôï 0, äåäïìÝíïõ üôé ôï ÷ùñßï åßíáé óõììåôñéêü ùò ðñïò ôïí y-Üîïíá.

ÌÜæá êáé êÝíôñï ìÜæáò.

Áò õðïèÝóïõìå üôé êÜèå óçìåßï ôïõ R Ý÷åé ðõêíüôçôá ρ[x,y].  Ôüôå ç ìÜæá Ì ôïõ ÷ùñßïõ åßíáé    

                                        M =∫ (∫_R)ρ[x, y] dxdy
                                        
Ôï êÝíôñï ìÜæáò ôïõ ÷ùñßïõ R óôï xy-åðßðåäï åßíáé ôï óçìåßï {x*,y*}, üðïõ

    x*= (1/M)∫ (∫_R)x  ρ[x, y] dxd y  êáé 
      y*= (1/M)∫ (∫_R)y  ρ[x, y] dxd y .

ÐáñÜäåéãìá 9.

Èåùñïýìå ôï ÷ùñßï  R óôï xy-åðßðåäï:

Rplot  = Plot[-x^2 + 4, {x, -2, 2}];

Õðïëïãßæïõìå ôï êÝíôñï ìÜæáò ôïõ ðáñáðÜíù ÷ùñßïõ üôáí ç ðõêíüôçôá åßíáé ρ[x,y]=(((x + y))^2).

Clear[ρ]
\(ρ[x_, y_] = \((x + y)\)\^\(\(2\)\(
\)\)\)
\(Clear[xstar, \ ystar, mass]
\)
\(mass = 
      Integrate[ρ[x, y], {x, \(-2\), 2}, {y, 0, 4 - x^2}]
\)
\(xstar = \(\(Integrate[x\ ρ[x, y], {x, \(-2\), 2}, {y, 0, 4 - x^2}]/
        mass\)\(
\)\)\)
\(ystar = \(\(Integrate[y\ ρ[x, y], {x, \(-2\), 2}, {y, 0, 4 - x^2}]/
        mass\)\(
\)\)\)

Óçìåéþíïõìå ôï êÝíôñï ìÝóá óôï ÷ùñßï R:

Clear[centroid]
centrofmass=Graphics[{Blue,PointSize[.025],Point[{xstar,ystar}]}];
Show[Rplot,centrofmass];

Ôï êÝíôñï ìÜæáò äåí Ý÷åé x-óõíôåôáãìÝíç ßóç ìå ôï 0. Ï ëüãïò åßíáé üôé ç óõíÜñôçóç ðõêíüôçôáò äåí åßíáé óõììåôñéêÞ ùò ðñïò ôïí y-Üîïíá.

ÐñïâëÞìáôá

Ðñüâëçìá 1 ( Õðïëïãéóìüò äéðëïý ïëïêëçñþìáôïò ìå ôï Mathematica)

a) Õðïëïãßóôå ôï ïëïêëÞñùìá (∫_0^4)(∫_0^2)f[x, y] dxdy  üðïõ  f[x,y] = 2+ (x^2) + (y^2)

b) Ôé õðïëïãßæåé ôï ðáñáðÜíù ïëïêëïêëÞñùìá?

Ðñüâëçìá 2 (¼ãêïò ìå ÷ñÞóç äéðëïý ïëïêëçñþìáôïò)

\(Ï\ ðáñáêÜôù\ êþäéêáò\ äßíåé\ ìéá\ ðáñáìåôñéêïðïßçóç\ ôïõ\ åëëåéøïåéäïýò\ x\^2\/4\) + ((y^2)/16) ≤ 1.

Clear[f,x,y,t,r]
x= 2r Cos[t]
y= 4r Sin[t]

ÊÜíïõìå ôçí ãñáöéêÞ ðáñÜóôáóç ôçò f[x, y]= 20 - (x^2) -(y^2) ìáæß ìå ôï åëëåéøïåéäÝò ((x^2)/4) + ((y^2)/16) ≤ 1.

f[x_, y_] = 20 - (x^2) - (y^2)
ParametricPlot3D[{{x, y, f[x, y]}, {x, y, 0}}, {t, 0, 2  π}, {r, 
          0, 1}];

Âñåßôå ôá üñéá ïëïêëÞñùóçò êáé óõìðëçñþóôå ôïí ðáñáêÜôù êþäéêá ãéá íá âñåßôå ôïí üãêï ôïõ óôåñåïý ðïõ öñÜóóåôáé ðñïò ôá ðÜíù áðü ôï ãñÜöçìá ôçò f[x,y] êáé ðñïò ôá êÜôù áðü ôï ÷ùñßï ((x^2)/4) + ((y^2)/16) ≤ 1 óôï  xy-åðßðåäï.

Clear[á, â, x, y, f]
f[x_, y_] = 20 - (x^2) - (y^2)

â[x_] = ??

\(á[x_] = \ \(\(??\)\(
\)\(\ \ \)\)\)

volume = ??

Ðñüâëçìá 3 (Åìâáäüí ÷ùñßïõ ìå ÷ñÞóç äéðëïý ïëïêëçñþìáôïò)

¸óôù R ôï ÷ùñßï ôçò ðáñáêÜôù ãñáöéêÞò ðáñÜóôáóçò

Clear[x, y];
\(\(ParametricPlot[{y\^2 - 4, y}, {y, \(-2\), 2}];\)
\)

a) Õðïëïãßóôå ôï äéðëü ïëïêëÞñùìá ôçò óõíÜñôçóçò f[x,y]= 1 åðß ôïõ R.

Clear[x, y, á, â, 
      f]
á[
        y_] = -2
â[y_] = 
      2
f[x_, y_] = 1;
Integrate[f[x, y], {y, -2, 2}, {x, (y^2) - 4, 0}]

b) Ôé ðáñéóôÜíåé ôï ðáñáðÜíù ïëïêëÞñùìá?

ÐáñÜñôçìá

ÃñáöéêÝò ðáñáóôÜóåéò  óõíáñôÞóåùí 2 ìåôáâëçôþí

Óå áõôÞí ôçí ðáñÜãñáöï äåß÷íïõìå ðùò íá êÜíïõìå ãñáöéêÝò ðáñáóôÜóåéò ðñáãìáôéêþí óõíáñôÞóåùí äýï ìåôáâëçôþí ìå ÷ñÞóç ôïõ Mathematica.

ÐáñÜäåéãìá 1

Ï ðáñáêÜôù êþäéêáò ó÷åäéÜæåé ôçí óõíÜñôçóç f[x,y]= 10 - (x^2) -(y^2) ãéÜ ôéò ôéìÝò  -2 ≤ x ≤ 2 êáé -2 ≤ y ≤ 2 êáèþò êáé ôï ÷ùñßï ôïõ xy-åðéðÝäïõ ãéá ôï ïðïßï -2 ≤ x ≤ 2 êáé -2 ≤ y ≤ 2.

\(Clear[f, x, y, gr1]
\)
\(\(f[x_, y_] = 10 - x\^2 - y\^2;\)
\)
\(\(gr1 = 
        ParametricPlot3D[\ {{x, y, 0}, {x, y, 
              f[x, y]}}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-2\), 
            2}, {y, \(-2\), 2}];\)\)

Ãéá íá ó÷åäéÜóïõìå Ýíá óçìåßï ðÜíù óôçí åðéöÜíåéá ðáßñíïõìå ðñþôá Ýíá óçìåßï {(Subscript[x, 0]),(Subscript[y, 0]),0} óôï äïèÝí ÷ùñßï óôï xy-åðßðåäï êáé ìåôÜ âñßóêïõìå ôï áíôßóôïé÷ï óçìåßï {(Subscript[x, 0]),(Subscript[y, 0]),f[(Subscript[x, 0]),(Subscript[y, 0])]} óôçí åðéöÜíåéá.  

Ãéá ðáñÜäåéãìá, áò ó÷åäéÜóïõìå ôï óçìåßï ôçò åðéöÜíåéáò ðïõ áíôéóôïé÷åß óôï óçìåßï ôïõ ÷ùñßïõ óôï xy-åðßðåäï ðïõ Ý÷åé óõíôåôáãìÝíåò {-1,-1,0}.

Clear[a,b,gr2];
a=-1;
b=-1;
gr2=Graphics3D[{Red,Thickness[.02],
Line[{ {a,b,0},{a,b,f[a,b]}}]}];
gr3=Graphics3D[{
 {Blue,PointSize[.06],Point[{a,b,0}]},
 {Blue,PointSize[.06],Point[{a,b,f[a,b]}]} }];
setup1=Show[gr1,gr2,gr3];

Ï ðáñáêÜôù êþäéêáò ó÷åäéÜæåé ôçí åéêüíá ôïõ óçìåßïõ {0,-.5,0}.

Clear[a,b,gr2];
a=0;
b=-.5;
gr2=Graphics3D[{Red,Thickness[.02],
Line[{ {a,b,0},{a,b,f[a,b]}}]}];
gr3=Graphics3D[{
 {Blue,PointSize[.06],Point[{a,b,0}]},
 {Blue,PointSize[.06],Point[{a,b,f[a,b]}]} }];
setup2=Show[gr1,gr2,gr3];

Ìðïñïýìå öõóéêÜ íá óõíäõÜæïõìå ãñáöéêÝò ðáñáóôÜóåéò:

Show[setup1,setup2];

ÐáñÜäåéãìá 2

Ôï ðáñáêÜôù åßíáé ôï ãñÜöçìá ôçò óõíÜñôçóçò f[x,y]= 3.1 (x^2)+ 2.3 (y^2) ìå ôéò åéêüíåò äéáöüñùí óçìåßùí ôïõ xy-åðéðÝäïõ.

Clear[f, x, k, gr1, gr2, gr3, gr4, n]
\(\(f[x_, y_] = \ 3.1\ x\^2 + \ 2.3\ y\^2;\)
\)
\(\(n = 100;\)
\)
\(\(rpoints\  = \ 
        Table[
{x[k] = Random[Real, {N[\(-2\)], N[3]}], 
\ 
            y[k] = Random[Real, {N[\(-1\)], N[4]}], 
\ f[x[k], y[k]]}, {k, 
            1, n}];\)\)
\(\(gr1 = 
        ParametricPlot3D[{{x, y, 0}, {x, y, 
              f[x, y]}}, 
\ \ \ \ \ {x, \(-2\), 3}, {y, \(-1\), 4}, 

          DisplayFunction -> Identity];\)

\)
\(\(gr2 = 

        Table[Graphics3D[{Red, 

              Line[{{x[k], y[k], 0}, {x[k], y[k], f[x[k], y[k]]}}]}], {k, 1, 
            n}];\)
\)
\(\(gr3 = \ 
        Table[Graphics3D[{PointSize[
                0.02], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Point[{x[k], 
                  y[k], 0}]}], {k, 1, n}];\)
\)
\(\(gr4 = \ 
        Table[Graphics3D[{PointSize[
                0.02], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Point[{x[k], 
                  y[k], f[x[k], y[k]]}]}], {k, 1, n}];\)\)
\(\(Show[gr1, gr2, gr3, gr4, 
ViewPoint -> CMView, 

        BoxRatios -> {2, 2, 1}, 

        DisplayFunction -> $DisplayFunction];\)
\)

Example 3

Óå áõôü ôï ðáñÜäåéãìá èåùñïýìå ìéá óõíÜñôçóç, ôçí f[x,y] = xy  ôçò ïðïßáò ç ãñáöéêÞ ðáñÜóôáóç äåí åßíáé ïëüêëçñç ðÜíù áðü ôï xy-åðßðåäï.

Clear[f,x,k,gr1,gr2,gr3,gr4,n]
f[x_,y_]= x y;

n=100;

rpoints = Table[
{x[k]=Random[Real,{N[-2],N[3]}],
 y[k]=Random[Real,{N[-1],N[4]}],
 f[x[k],y[k]]},{k,1,n}];
gr1=ParametricPlot3D[{{x,y,0},{x,y,f[x,y]}},
     {x,-2,3},{y,-1,4},
DisplayFunction->Identity];


gr2=
Table[Graphics3D[{Red,
Line[{{x[k],y[k],0},{x[k],y[k],f[x[k],y[k]]}}]}],{k,1,n}];

gr3= Table[Graphics3D[{PointSize[0.02],
                   Point[{x[k],y[k],0}]}],{k,1,n}];

gr4= Table[Graphics3D[{PointSize[0.02],
                   Point[{x[k],y[k],f[x[k],y[k]]}]}],{k,1,n}];
Show[gr1,gr2,gr3,gr4,
DisplayFunction->$DisplayFunction,
PlotRange->All,BoxRatios->{1,1,2}];

Ôá óçìåßá ôçò åðéöÜíåéáò ðïõ åßíáé ðÜíù áðü ôï xy-åðßðåäï áíôéóôïé÷ïýí óå óçìåßá {x,y,0} üðïõ ôï f[x,y] åßíáé èåôéêüò áñéèìüò.
Ôá óçìåßá ôçò åðéöÜíåéáò ðïõ åßíáé êÜôù áðü ôï xy-åðßðåäï áíôéóôïé÷ïýí óå óçìåßá {x,y,0} üðïõ ôï f[x,y] åßíáé áñíçôéêüò áñéèìüò.
Ôá óçìåßá ôçò åðéöÜíåéáò åðß ôïõ xy-åðéðÝäïõ áíôéóôïé÷ïýí óå óçìåßá {x,y,0} üðïõ f[x,y] = 0.