ÔñéðëÜ ïëïêëçñþìáôá

◆Ðïõ ÷ñçóéìïðïéïýíôáé ôá ôñéðëÜ
      ïëïêëçñþìáôá ;

Áêñéâþò üðùò ôá äéðëÜ ïëïêëçñþìáôá ÷ñçóéìïðïéÞèçêáí ãéá ôïí õðïëïãéóìü åìâáäþí, ôá ôñéðëÜ ïëïêëçñþìáôá ìðïñïýí íá ÷ñçóéìïðïéçèïýí ãéá ôïí õðïëïãéóìü ôïõ 
üãêïõ åíüò 3-äéÜóôáôïõ ÷ùñßïõ
×ñçóéìåýïõí åðßóçò ãéá ôçí åýñåóç ôçò ìÝóçò ôéìÞò ìéáò óõíÜñôçóçò ïñéóìÝíçò óå Ýíá 3-äéÜóôáôï ÷ùñßï êáèþò åðßóçò ôçò ìÜæáò, ôùí óõíôåôáãìÝíùí ôïõ êÝíôñïõ ìÜæáò êáé ôçò ñïðÞò áäñáíåßáò 3-äéÜóôáôùí óùìÜôùí.

◆Ïñéóìüò ôñéðëïý ïëïêëçñþìáôïò

¼ðùò óôá äéðëÜ ïëïêëçñþìáôá, îåêéíÞóáìå ìå ïëïêëÞñùóç ïñèïãùíßùí ðáñáëëçëïãñÜììùí, óôá ôñéðëÜ ïëïêëçñþìáôá ôï áðëïýóôåñï ÷ùñßï åðß ôïõ ïðïßïõ áðïäåéêíýåôáé ç ýðáñîç ïëïêëçñþìáôïò ãéá ìéá óõíå÷Þ óõíÜñôçóç f åßíáé ôï ïñèïãþíéï ðáñáëëçëåðßðåäï, äçëáäÞ, Ýíá óôåñåü ìå  ðëåõñÝò  ðáñÜëëçëåò óôïõò Üîïíåò x, y êáé z.  ÓõãêåêñéììÝíá, 
                     R = [(Subscript[a, 1]), (Subscript[b, 1])]×[(Subscript[a, 2]), 
            (Subscript[b, 2])]×[(Subscript[a, 3]), (Subscript[b, 3])] 
êáé ï üãêïò ôïõ R  åßíáé 
                    Vol(
          R) = ((Subscript[b, 1]) - (Subscript[a, 1])) ((Subscript[b, 2]) - (Subscript[a, 2])) ((Subscript[b, 3]) - (Subscript[a, 3]))
                    
Ìßá äéáìÝñéóç P ôïõ R áðáñôßæåôáé áðü äéáìåñßóåéò (Subscript[P, 1]),(Subscript[P, 2]),(Subscript[P, 3]) ôùí äéáóôçìÜôùí [(Subscript[a, 1]), (Subscript[b, 1])], [(Subscript[a, 2]), (Subscript[b, 2])], [(Subscript[a, 3]), 
        (Subscript[b, 3])] áíôßóôïé÷á êáé ÷ùñßæåé ôï ïñèïãþíéï R óå ìéêñÜ õðï-ïñèïãþíéá ôá ïðïßá ôá óõìâïëßæïõìå ìå ôï ãñÜììá S. ¼ðùò áêñéâþò êÜíáìå óôá äéðëÜ ïëïêëçñþìáôá, èåùñïýìå Üíù êáé êÜôù áèñïßóìáôá ùò åîÞò:
                L(P, f) = (∑_S)(inf f) Vol(S)  
êáé
               U(P, f) = (∑_S)(sup f) Vol(S).

Ìßá óõíÜñôçóç f èá ëÝãåôáé ïëïêëçñþóéìç åðß ôïõ ïñèïãùíßïõ  R  åÜí õðÜñ÷åé ìïíáäéêüò ðñáãìáôéêüò áñéèìüò  Á  ôÝôïéïò þóôå ãéá êÜèå äýï äéáìåñßóåéò  P, Q ôïõ  R  íá éó÷ýåé 
                  L(P, f) ≤ A ≤ U(Q, f)
                  
Èåþñçìá: ¸óôù R ïñèïãþíéï êáé  f : R->R öñáãìÝíç óõíÜñôçóç ç ïðïßá åßíáé óõíå÷Þò óôï R  åêôüò ßóùò áðü óçìåßá ôá ïðïßá áíÞêïõí óå ðåðåñáóìÝíåò ôï ðëÞèïò äéáöïñßóéìåò åðéöÜíåéåò óôï R. Ôüôå ç f åßíáé ïëïêëçñþóéìç.
Óõìâïëéóìüò: ∫∫(∫_R) 
              f =\(TraditionalForm\`∫\(∫\(∫\_R\ \(f(x, y, 
                z)\) dx dy dz\)\)\).

Åðåêôåßíïõìå ôçí ïêëïêëÞñùóç óå öñáãìÝíá ÷ùñßá  Ó  ãåíéêüôåñá ôïõ ïñèïãùíßïõ ðáñáëëçëåðéðÝäïõ  R  ùò åîÞò: èåùñïýìå ðáñáëëçëåðßðåäï  R ðïõ íá ðåñéÝ÷åé ôï Ó êáé èÝôïõìå
             f(x) = 0 ãéá   x ϵ R \ Ó êáé \(TraditionalForm\`∫\(∫\(\(∫\_Ó\)\(\ \)\(f\)\(\ \)\)\)\)=\(TraditionalForm\`∫\(∫\(\(∫\_R\)\(\ \)\(f\)\(\ \)\)\)\)

◆ÏëïêëÞñùóç åðß ïñèïãùíßïõ 
      ðáñáëëçëåðéðÝäïõ (õðïëïãéóìüò)

Ãéá ôïí õðïëïãéóìü ôùí ôñéðëþí ïëïêëçñùìÜôùí ÷ñçóéìïðïéïýìå ôï Èåþñçìá Fubini äéáôõðùìÝíï óå 3 äéáóôÜóåéò. Ìå áõôü ôïí ôñüðï ï õðïëïãéóìüò åíüò ôñéðëïý ïëïêëçñþìáôïò áíÜãåôáé óå äéáäï÷éêÝò ïëïêëçñþóåéò Riemann ðñáãìáôéêþí óõíáñôÞóåùí.

ÐáñÜäåéãìá

Õðïëïãéóìüò ôïõ ôñéðëïý ïëïêëçñþìáôïò
 \(TraditionalForm\`∫\(∫∫\_\(\(\ \)\(B\)\)\)\) (x^2) (y^3) z dx dy dz   
 üðïõ B ðáñáëëçëåðßðåäï ìå  0 ≤ x ≤ 2; -1 ≤ y ≤ 3 ; 4 ≤ z ≤ 7.

×ñçóéìïðïéþíôáò ôï Mathematica Ý÷ïõìå ôçí ðáñáêÜôù ôñéóäéÜóôáôç ãñáöéêÞ ðáñÜóôáóç ôïõ ðáñáëëçëåðéðÝäïõ.

Clear[setup1]
setup1 = 
        Show[box[{0, 2}, {-1, 3}, {4, 7}], Axes -> True, 
          AxesLabel -> {"x", "y", "z"}];

×ñçóéìïðïéþíôáò êáé ðÜëé ôï Mathematica  ìðïñïýìå íá õðïëïãßóïõìå ôï ïëïêëÞñùìá áìÝóùò. Ôï ìüíï ðïõ ÷ñåéÜæåôáé íá êÜíïõìå åßíáé íá êáèïñßóïõìå ôá üñéá ôçò ïëïêëÞñùóçò (ôá ïðïßá ìáò äßíïíôáé).

Clear[x, y, z]
(∫_4^7)(∫_(-1)^3)(∫_0^2) (x^2) (y^3) z \[DifferentialD]x \[DifferentialD]y \[DifferentialD]z
Integrate[
      (x^2) (y^3) z,  {z,  4,  7},  {y,  -1,  3},  {x,  0,  
        2}]

Ï õðïëïãéóìüò ãßíåôáé óå ôñßá âÞìáôá ÷ñçóéìïðïéþíôáò ôï Èåþñçìá Fubini. ÊÜèå âÞìá åßíáé Ýíá áðëü ïëïêëÞñùìá. 
Ðñþôï âÞìá:

Clear[first, second, third, x, y, z, f]
f[x_, y_, z_] = (x^2) (y^3) z;
first[y_, 
        z_] = (∫_0^2) f[x, y, z] \[DifferentialD]x
\(		Integrate[f[x, \ y, \ z], \ {x, \ 0, \ 2}]\)

Äåýôåñï âÞìá,

second[
        z_] = (∫_(-1)^3) first[y, z] \[DifferentialD]y
\(		Integrate[first[y, \ z], \ {y, \ \(-1\), \ 3}]\)

Êáé ôï ôñßôï âÞìá:

third = (∫_4^7) second[z] \[DifferentialD]z
\(		Integrate[second[z], \ {z, \ 4, \ 7}]\)

Èá åîåôÜóïõìå êÜèå Ýíá áðü ôá ôñßá âÞìáôá ÷ùñéóôÜ.
Ôé âñÞêáìå üôáí õðïëïãßóáìå ôï ðñþôï ïëïêëÞñùìá, äçëáäÞ, üôáí õðïëïãßóáìå ôçí óõíÜñôçóç first[y,z] ?

Clear[setup2]
setup2[y0_, z0_] := 
        Graphics3D[{Blue, Thickness[ .02], 
            Line[{{0, y0, z0}, {2, y0, z0}}]}];
firstintegral = Show[setup1, setup2[1, 5]];

AðÜíôçóç: Õðïëïãßóáìå ôï ïëïêëÞñùìá ôçò óõíÜñôçóçò 
                   (x^2) (y^3) z 
åðß  åõèõãñÜììïõ ôìÞìáôïò ðïõ õðïäåéêíýåôáé óôçí ðáñáðÜíù ãñáöéêÞ ðáñÜóôáóç.
Åäþ ôï y êáé ôï z åßíáé óôáèåñÜ, äçëáäÞ èåùñïýìå ôçí
 (x^2) (y^3) z  ùò óõíÜñôçóç ìüíï ôïõ x.

Ôé êÜíáìå üôáí õðïëïãßóáìå ôï äåýôåñï ïëïêëÞñùìá, äçëáäÞ, üôáí õðïëïãßóáìå ôçí óõíÜñôçóç second[z] ?

Clear[setup2, strip]
step =  .4;
setup2[y0_, z0_] := 
        Graphics3D[{Blue, Thickness[ .02], 
            Line[{{0, y0, z0}, {2, y0, z0}}]}];
strip[z0_] := 
        Table[setup2[y0, z0], {y0, -1, 3, step}];
Show[setup1, strip[5]];

AðÜíôçóç: Ïëïêëçñþóáìå åðß ìßáò ëùñßäáò ðïõ õðïäåéêíýåôáé óôçí ðáñáðÜíù ãñáöéêÞ ðáñÜóôáóç. 
Åäþ êñáôÞóáìå ôï z óôáèåñü êáé ç ìáôáâëçôÞ y êéíåßôï áðü -1 Ýùò 3.

Óôçí ðáñáðÜíù ãñáöéêÞ ðáñÜóôáóç ÷ñçóéìïðïéÞóáìå äéÜöïñåò äéáêñéôÝò ôéìÝò ãéÜ  ôï y. ÅÜí ðÜñïõìå ìéêñüôåñï "âÞìá" ãéá ôéò ôéìÝò ôïõ y, ôüôå ç ãñáöéêÞ ðáñÜóôáóç èá öáßíåôáé ùò åðéöÜíåéá.

Clear[setup2, strip]
step =  .1;
setup2[y0_, z0_] := 
        Graphics3D[{Blue, Thickness[ .02], 
            Line[{{0, y0, z0}, {2, y0, z0}}]}];
strip[z0_] := 
        Table[setup2[y0, z0], {y0, -1, 3, step}];
secondintegral = Show[setup1, strip[5]];

Ôé êÜíáìå üôáí õðïëïãßóáìå ôï ôñßôï ïëïêëÞñùìá third ?

Clear[setup3, strips, stepz]
stepz =  .5;
setup3[z0_] := 
        Graphics3D[
          Polygon[{{0, -1, z0}, {2, -1, z0}, {2, 3, z0}, {0, 3, 
                z0}, {0, -1, z0}}]];
strips = Table[setup3[z0], {z0, 4, 7, stepz}];
Show[setup1, strips];

ÁðÜíôçóç: ÁöÞíïíôáò ôçí ìåôáâëçôÞ z íá êéíåßôáé áðü 4 óôï 7, ç ëùñßäá ðïõ åßäáìå ðáñáðÜíù óÜñùóå ïëüêëçñï ôï ðáñáëëçëåëßðåäï.

Èåùñþíôáò ìéêñü "âÞìá" ãéá ôéò ôéìÝò ôïõ z (i.e. stepz=.1) âëÝðïõìå ôé ãßíåôáé ìå ôçí ïëïêëÞñùóç.

Clear[setup3, strips, stepz]
stepz =  .1;
setup3[z0_] := 
        Graphics3D[
          Polygon[{{0, -1, z0}, {2, -1, z0}, {2, 3, z0}, {0, 3, 
                z0}, {0, -1, z0}}]];
strips = Table[setup3[z0], {z0, 4, 7, stepz}];
thirdintegral = Show[setup1, strips];

Ôá ðáñáðÜíù âÞìáôá óå ôáéíßåò ìéêñïý ìÞêïõò.
ÂÞìá 1: y, z óôáèåñÜ

Clear[line, r, step]
step =  .25;
line[x_] := 
      Graphics3D[{Blue, Thickness[ .02], 
          Line[{{0, 1, 5}, {x, 1, 5}}]}]
Table[Show[setup1, line[x]];, {x, 0, 2, step}];

ÂÞìá 2: z óôáèåñü

Clear[setup2, strip, polygon]
step =  .25;
polygon[r_, z0_] := 
        Graphics3D[
          Polygon[{{0, -1, z0}, {2, -1, z0}, {2, -1 + r, 
                z0}, {0, -1 + r, z0}, {0, -1, z0}}]];
Table[Show[setup1, polygon[r, 5]];, {r, 0, 4, 
          step}];

ÂÞìá 3: ôßðïôå óôáèåñü (x, y êáé z åßíáé ìåôáâëçôÝò).

Clear[stepz]
stepz =  .5;
Table[Show[setup1, setup3[z0]];, {z0, 4, 7, 
          stepz}];

◆ÔñéðëÜ ïëïêëçñþìáôá óôåñåþí: 
      êáèïñéóìüò ïñßùí ïëïêëÞñùóçò  
      êáé õðïëïãéóìïß

ÏëïêëÞñùóç åðß óôåñåïý ðïõ öñÜóóåôáé áðü åðéöÜíåéåò 
z= (Subscript[g, 1])[x,y] êáé z = (Subscript[g, 2])[x,y],  
üðïõ  x êáé y áíÞêïõí óå êÜðïéï ÷ùñßï ôïõ xy-åðéðÝäïõ.

ÐáñÜäåéãìá 
Õðïëïãßæïõìå ôï ôñéðëü ïëïêëÞñùìá 
                  \(TraditionalForm\`∫\(∫∫\_\(\(\ \)\(R\)\)\)\)  (x^2) (y^2) z dz dx dy, 
 üðïõ R åßíáé ôï ÷ùñßï ðïõ öñÜóóåôáé áðü ôéò åðéöÜíåéåò
 (Subscript[g, 1])[x,y]= 6 -((x^2)+ (y^2)) êáé (Subscript[g, 2])[x,y] = ((x^2)+ (y^2))-2.

Åßíáé ðÜíôá ÷ñÞóéìï íá Ý÷ïõìå óôï ìõáëü ìáò ìßá åéêüíá ôïõ óôåñåïý åðß ôïõ ïðïßïõ ïëïêëçñþíïõìå. 
Ôï Mathematica åßíáé ÷ñÞóéìï åñãáëåßï ãéá ãñáöéêÝò ðáñáóôÜóåéò.

Óôçí ðáñáêÜôù ãñáöéêÞ ðáñÜóôáóç ðÞñáìå Ýíáí êýêëï áêôßíáò 1,5 (ç áêôßíá äéáëÝ÷èçêå ôõ÷áßá, áëëÜ ç åðéëïãÞ ó÷Þìáôïò êýêëïõ äåí åßíáé ôõ÷áßá) êáé æçôÜìå áðü ôï  Mathematica íá ó÷åäéÜóåé ôá ôìÞìáôá ôùí åðéöáíåéþí  (Subscript[g, 1])[x,y] êáé (Subscript[g, 2])[x,y] ðïõ âñßóêïíôáé ðÜíù áðü áõôü ôï êýêëï.

Clear[x, y, z, g1, g2, r, t]
g1[x_, y_] = 6 - ((x^2) + (y^2));
g2[x_, y_] = ((x^2) + (y^2)) - 2;
x[r_, t_] = r Cos[t];
y[r_, t_] = r Sin[t];
ParametricPlot3D[
        Evaluate[{{x[r, t], y[r, t], g1[x[r, t], y[r, t]]}, {x[r, t], 
              y[r, t], g2[x[r, t], y[r, t]]}}], {t, 0, 2 π}, {r, 0, 
          1.5}];

ÊÜíïõìå ôçí ßäéá ãñáöéêÞ ðáñÜóôáóç åðéëÝãïíôáò ôçí áêôßíá  0≤r≤2 (áõôÞ ç åðéëïãÞ áêôßíáò äåí åßíáé ôõ÷áßá äéüôé 
6 -((x^2)+ (y^2)) = ((x^2)+ (y^2))-2 ⟹ 8 = 2 ((x^2)+ (y^2))  ⟹  4 = (x^2)+ (y^2) 
ðïõ åßíáé êýêëïò áêôßíáò 2).

Clear[x, y, z, g1, g2, r, t]
g1[x_, y_] = 6 - ((x^2) + (y^2));
g2[x_, y_] = ((x^2) + (y^2)) - 2;
x[r_, t_] = r Cos[t];
y[r_, t_] = r Sin[t];
ParametricPlot3D[{ {x[r, t], y[r, t], 
            g1[x[r, t], y[r, t]]}, {x[r, t], y[r, t], 
            g2[x[r, t], y[r, t]]}}, {t, 0, 2 π}, {r, 0, 
          2}];

Ïé äýï åðéöÜíåéò ôÝìíïíôáé üôáí r = 2.  Áõôü óçìáßíåé üôé ïé óõíôåôáãìÝíåò x êáé y (ôùí óçìåßùí ôïõ óôåñåïý) ðñÝðåé íá âñßóêïíôáé ìÝóá óôïí êýêëï êÝíôñïõ 0 êáé áêôßíáò 2 óôï xy-åðßðåäï.

Èåùñïýìå ôá x êáé y óôáèåñÜ.  Ôüôå ôï z äéáôñÝ÷åé ôï äéÜóôçìá áðü ôï (Subscript[g, 1])[x,y] óôï (Subscript[g, 2])[x,y].  Ïëïêëçñþíïíôáò ùò ðñïò z, ôï ôñéðëü ïëïêëÞñùìá áíÜãåôáé óå Ýíá äéðëü ïëïêëÞñùìá åðß ôïõ êýêëïõ áêôßíáò 2 óôï xy-åðßðåäï.

Ï õðïëïãéóìüò ôïõ ïëïêëçñþìáôïò (ìå ÷ñÞóç ôïõ
Mathematica ãéá ôïõò õðïëïãéóìïýò). 

Ðñþôá ïëïêëçñþíïõìå ùò ðñïò z ìå üñéá ïëïêëÞñùóçò  (Subscript[g, 1]) [x, y] and (Subscript[g, 2]) [x, y].

Clear[first, second, third, x, y, z, f]
f[x_, y_, z_] = (x^2) (y^2) z;
first[y_, x_] = 
      Expand[(∫_(g2[x, y])^(g1[x, y]))f[x, y, 
              z] \[DifferentialD]z]

Óôçí óõíÝ÷åéá ïëïêëçñþíïõìå óôïí êýêëï óôï xy-åðßðåäï.  ÊñáôÜìå ôï x óôáèåñü êáé ïëïêëçñþíïõìå ùò ðñïò y ìå üñéá ïëïêëÞñùóçò -(Sqrt[4 - (x^2)]) êáé (Sqrt[4 - (x^2)]).

second[
        x_] = (∫_(-(Sqrt[4 - (x^2)]))^((Sqrt[4 - (x^2)])))first[y, 
            x] \[DifferentialD]y

ÔÝëïò ïëïêëçñþíïõìå ùò ðñïò x.

third = (∫_(-2)^2) second[x] \[DifferentialD]x

ÅëÝã÷ïõìå åáí ç åöáñìïãÞ ìáò äïõëåýåé óùóôÜ:

(∫_(-2)^2)((∫_(-(Sqrt[4 - (x^2)]))^((Sqrt[4 - (x^2)])))((∫_(g2[x, y])^(g1[x, y]))f[x, y, 
                      z] \[DifferentialD]z) \[DifferentialD]y) \[DifferentialD]x

ÏëïêëÞñùóç åðß óôåñåïý ðïõ öñÜóóåôáé áðü åðéöÜíåéåò 
y = (Subscript[g, 1])[x,z] êáé y = (Subscript[g, 2])[x,z],  
üðïõ  x êáé z áíÞêïõí óå êÜðïéï ÷ùñßï ôïõ xz-åðéðÝäïõ.

Õðïëïãßæïõìå ôï ôñéðëü ïëïêëÞñùìá 
                   ∫∫(∫_R) (x^4) y (z^8) dz dx dy
 üðïõ R åßíáé ôï ÷ùñßï ðïõ öñÜóóåôáé áðü ôéò åðéöÜíåéåò
y= ((x^2)+(z^2)) êáé y = 18 - ((x^2)+(z^2)).

Âñßóêïõìå ôçí ôïìÞ ôùí äýï åðéöáíåéþí üðùò ðñïçãïõìÝíùò, äçëáäÞ,  ëýíïíôáò ôçí åîßóùóç
          ((x^2)+(z^2)) = 18 - ((x^2)+(z^2)) 
 ç ïðïßá ìáò äßíåé (x^2)+(z^2) = 9.  
Áõôü óçìáßíåé üôé ïé óõíôåôáãìÝíåò x êáé z (ôùí óçìåßùí ôïõ óôåñåïý) ðñÝðåé íá âñßóêïíôáé ìÝóá óôïí êýêëï êÝíôñïõ 0 êáé áêôßíáò 3 óôï xz-åðßðåäï.

ÊÜíïõìå ôçí ãñáöéêÞ ðáñÜóôáóç ìå åðéëïãÞ áêôßíáò (óôéò ðáñáìåôñéêÝò åîéóþóåéò) r=2.

Clear[x, y, z, g1, g2, r, t]
g1[x_, z_] = (x^2) + (z^2);
g2[x_, z_] = 18 - ((x^2) + (z^2));
x[r_, t_] = r Cos[t];
z[r_, t_] = r Sin[t];
ParametricPlot3D[
        Evaluate[{{x[r, t], g1[x[r, t], z[r, t]], z[r, t]}, {x[r, t], 
              g2[x[r, t], z[r, t]], z[r, t]}}], {t, 0, 2 π}, {r, 0, 2}, 
        AxesLabel -> {"x", "y", "z"}];

ÊÜíïõìå ôþñá ôçí ãñáöéêÞ ðáñÜóôáóç ìå áêôßíá óôéò ðáñáìåôñéêÝò åîéóþóåéò r=3 ðïõ âñÞêáìå üôé õëïðïéåß ôçí ôïìÞ ôùí äýï åðéöáíåéþí.

Clear[x, y, z, g1, g2, r, t]
g1[x_, z_] = (x^2) + (z^2);
g2[x_, z_] = 18 - ((x^2) + (z^2));
x[r_, t_] = r Cos[t];
z[r_, t_] = r Sin[t];
ParametricPlot3D[
        Evaluate[{{x[r, t], g1[x[r, t], z[r, t]], z[r, t]}, {x[r, t], 
              g2[x[r, t], z[r, t]], z[r, t]}}], {t, 0, 2 π}, {r, 0, 3}, 
        AxesLabel -> {"x", "y", "z"}];

Ï õðïëïãéóìüò ôïõ ïëïêëçñþìáôïò (ìå ÷ñÞóç ôïõ
Mathematica). 

Ðñþôá ïëïêëçñþíïõìå ùò ðñïò y ìå üñéá ïëïêëÞñùóçò  (Subscript[g, 1]) [x, z] êáé (Subscript[g, 2]) [x, z].

Clear[first, second, third, x, y, z, f]
f[x_, y_, z_] = (x^4) y (z^8);
first[x_, z_] = 
      Expand[(∫_(g1[x, z])^(g2[x, z]))f[x, y, 
              z] \[DifferentialD]y]

Óôçí óõíÝ÷åéá äïõëåýïõìå óôï xz-åðßðåäï, êñáôþíôáò ôï x óôáèåñü êáé ïëïêëçñþíïíôáò ùò ðñïò z.

second[
        x_] = (∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))first[x, 
            z] \[DifferentialD]z

ÔÝëïò ïëïêëçñþíïõìå ùò ðñïò x.

third = (∫_(-3)^3) second[x] \[DifferentialD]x

Ôá âÞìáôá óôçí ðáñáðÜíù ïëïêëÞñùóç Ý÷ïõí êáèáñÜ åðåîçãçìáôéêü ñüëï. ÅÜí èÝëïõìå, ïëüêëçñïò ï ðáñáðÜíù õðïëïãéóìüò ìðïñåß íá ãßíåé ìå ÷ñÞóç ôïõ  Mathematica  áð' åõèåßáò.

(∫_(-3)^3)((∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))((∫_(g1[x, z])^(g2[x, z]))f[x, y, 
                      z] \[DifferentialD]y) \[DifferentialD]z) \[DifferentialD]x

ÏëïêëÞñùóç åðß óôåñåïý ðïõ ðåñéãñÜöåôáé áðü áíéóüôçôåò 
a ≤ x ≤ b, (Subscript[g, 1])[x] ≤ y ≤ (Subscript[g, 2])[x] êáé
(Subscript[h, 1])[x,y] ≤ z ≤ (Subscript[h, 2])[x,y]

Ç ïëïêëÞñùóç åðß åíüò óôåñåïý Á ôï ïðïßï áðïôåëåßôáé áðü üëá ôá (x,y,z)  ãéá ôá ïðïßá éó÷ýïõí ïé ðáñáðÜíù áíéóüôçôåò ãéá êÜðïéåò äéáöïñßóéìåò óõíáñôÞóåéò 
 
 (Subscript[g, 1]), (Subscript[g, 2]): [a,b]->R ìå (Subscript[g, 1])(x) ≤ (Subscript[g, 2])(x) êáé 
 (Subscript[h, 1]), (Subscript[h, 2]): {(x,y) : a ≤ x ≤ b êáé (Subscript[g, 1])(x) ≤ y ≤ (Subscript[g, 2])(x)}->R ìå 
                     (Subscript[h, 1])(x,y) ≤ (Subscript[h, 2])(x,y)
 
 áíÜãåôáé óôïí õðïëïãéóìü ôïõ ðáñáêÜôù ïëïêëçñþìáôïò
 
        \(TraditionalForm\`∫\(∫∫\_\(\(\ \)\(Á\)\)\)\) f = (∫_a^b)((∫_((Subscript[g, 1])[x])^((Subscript[g, 2])[x]))((∫_((Subscript[h, 1])[x, y])^((Subscript[h, 2])[x, y]))f(
                          x, y, z) dz)dy)dx
 
 ÁõôÞ ç ðåñßðôùóç åìðåñéÝ÷åôáé óôéò ðñïçãïýìåíåò

ÏëïêëÞñùóç åðß óôåñåïý ðïõ ðåñéãñÜöåôáé áðü äýï áíéóüôçôåò: åýñåóç üãêïõ ôïõ óôåñåïý ðïõ êáèïñßæåôáé áðü ôçí ôïìÞ ôùí óöáéñþí (x^2) + (y^2) + (z^2) = (4^2) êáé (((x - 3))^2) + (y^2) + (z^2) = (2^2).

Áíáæçôïýìå ôïí üãêï ôïõ óôåñåïý Á ôï ïðïßï áðïôåëåßôáé áðü üëá ôá (x,y,z)  ãéá ôá ïðïßá éó÷ýïõí ïé áíéóüôçôåò 
   (x^2) + (y^2) + (z^2) ≤ (4^2) êáé (((x - 3))^2) + (y^2) + (z^2) ≤ (2^2)
Õðïëïãßæïõìå ôï ÷ùñßï R ôïõ xy-åðéðÝäïõ ðïõ ðåñéÝ÷åôáé óôåñåü Á:
{({{(x^2) + (y^2) = (4^2)}, {(((x - 3))^2) + (y^2) = (2^2)}})}⟹ ⋯ ⟹ {({{x = 21/6}, {y = ±(Sqrt[(4^2) - (((21/6))^2)])}})}
Óõíåðþò ôï ÷ùñßï R áðïôåëåßôáé áðü ôá ÷ùñßá  R = (Subscript[R, 1]) ⋃ (Subscript[R, 2]) üðïõ 
(Subscript[R, 1]) =  {(x, 
                y) │  1 ≤ x ≤ 
            21/6 ,  -(Sqrt[(2^2) - (((x - 3))^2)]) ≤ 
            y ≤ (Sqrt[(2^2) - (((x - 3))^2)])}
  êáé 
(Subscript[R, 2]) =  {(x, y) │ 21/
                  6 ≤ x ≤ 
              4 ,  -(Sqrt[(4^2) - (x^2)]) ≤ 
              y ≤ (Sqrt[(4^2) - (x^2)])}.

Ç ãñáöéêÞ ðáñÜóôáóç ôçò ïñéæüíôéáò ôïìÞò óôï ýøïò z = 0 (äçë. ôïõ xy-åðéðÝäïõ)

Clear[v, w, s1, s2,  c1,  c2,  t1,  t2,  t3]
s1 = 
        ParametricPlot[{4  Cos[w] ,  4  Sin[w]}, {w, -Pi/2, Pi/2},  
          DisplayFunction  ->  Identity];
s2 = 
        ParametricPlot[{3 + 2  Cos[v] ,  2  Sin[v]}, {v, Pi/3, 7  Pi/4},  
          DisplayFunction  ->  Identity];
Show[s1, s2,  
      DisplayFunction  ->  $DisplayFunction]

Ôï óôåñåü  Á ìðïñïýìå íá ôï "äïýìå" íá ó÷çìáôßæåôáé áöïý ðñïçãïõìÝíùò  ðåñéïñéóôïýìå óå êáôÜëëçëá ôìÞìáôá ôùí åìðëåêïìÝíùí óöáéñþí:

Clear[t, u, sph1, sph2]
sph1 = 
        ParametricPlot3D[{4  Cos[t] Cos[u], 4  Sin[t] Cos[u],  
            4  Sin[u]}, {t, 0, 5  Pi/3}, {u,  -Pi/3,  Pi/3},  
          DisplayFunction  ->  Identity];
sph2 = 
        ParametricPlot3D[{3 + 2  Cos[t] Cos[u], 2  Sin[t] Cos[u],  
            2  Sin[u]}, {t, Pi/6, 6  Pi/4}, {u,  -Pi/2,  Pi/2}, 
          DisplayFunction  ->  Identity];
Show[sph1,  sph2, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}, Boxed -> False,  
        DisplayFunction  -> $DisplayFunction ];

ÔÝëïò ï üãêïò ôïõ óôåñåïý Á äßíåôáé áðü ôï æçôïýìåíï ïëïêëÞñùìá

(∫_1^((21/6)))((∫_(-(Sqrt[(2^2) - (((x - 3))^2)]))^((Sqrt[(2^2) - (((x - 3))^2)])))((∫_(-(Sqrt[(2^2) - (y^2) - (((x - 3))^2)]))^((Sqrt[(2^2) - (y^2) - (((x - 3))^2)]))) 
                    1 \[DifferentialD]z) \[DifferentialD]y) \[DifferentialD]x+(∫_((21/6))^4)((∫_(-(Sqrt[(4^2) - (x^2)]))^((Sqrt[(4^2) - (x^2)])))((∫_(-(Sqrt[(4^2) - (y^2) - (x^2)]))^((Sqrt[(4^2) - (y^2) - (x^2)]))) 1 \[DifferentialD]z) \[DifferentialD]y) \[DifferentialD]x

◆ÁëëáãÞ ôçò óåéñÜò ïëïêëçñþóåùò

Äßíåôáé ôï ôñéðëü ïëïêëÞñùìá  \(∫\_0\%1\(∫\_0\%z\(∫\_0\%y\ \ xy\^2\ z\^3\)\)\) dx dy dz 
êáé Ýóôù üôé åðéèõìïýìå íá áëëÜîïõìå ôçí óåéñÜ ïëïêëÞñùóçò þóôå íá åßíáé dx dz dy.

Õðïëïãßæïõìå ôï ïëïêëÞñùìá (êÜôé ôÝôïéï, åí ãÝíåé, äåí ÷ñåéÜæåôáé)

Clear[x, y, z]
(∫_0^1)(∫_0^z)(∫_0^y) 
              x (y^2) (z^3) \[DifferentialD]x \[DifferentialD]y \[DifferentialD]z

ÊÜíïõìå ôçí ãñáöéêÞ ðáñÜóôáóç ôïõ óôåñåïý åðß ôïõ ïðïßïõ ïëïêëçñþíïõìå (áõôü åßíáé ðÜíôá ÷ñÞóéìï ü÷é üìùò êáé áðáñáßôçôï).

Clear[setup1, setupyz]
setupyz = 
        Graphics3D[{Red, Thickness[ .02], 
            Line[{{0, 0, 0}, {0, 0, 1}, {0, 1, 1}, {0, 0, 0}}]}];
setup1 = 
        Graphics3D[{Blue, Thickness[ .02], 
            Line[{{0, 0, 0}, {1, 1, 1}, {0, 0, 1}}]}];
setup2 = 
        Graphics3D[{Blue, Thickness[ .02], 
            Line[{{1, 1, 1}, {0, 1, 1}}]}];
Show[setupyz, setup1, setup2, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}, 
        Boxed -> False];

Ôï óôåñåü åðß ôïõ ïðïßïõ ïëïêëçñþíïõìå åßíáé ôï ðáñáðÜíù ôåôñÜåäñï ðïõ Ý÷åé âÜóç óôï yz-åðßðåäï (êüêêéíï ôñßãùíï). Ìðïñïýìå, åíáëáêôéêÜ, íá èåùñÞóïõìå üôé ç âÜóç ôïõ âñßóêåôáé óôï åðßðåäï y=x, Þ óôï åðßðåäï y=z.

Tï äïèÝí ïëïêëÞñùìá (∫_0^1)(∫_0^z)(∫_0^y) 
              (y^2) (z^3)dx dy dz õðïëïãßæåôáé ìå ôá åîÞò âÞìáôá (ùò ðñïò ôçí óåéñÜ ïëïêëÞñùóçò dx dy dz).

Clear[strip, setup3, step]
step =  .6;
strip[t_] := 
        Graphics3D[
          Polygon[{{0, 0, t}, {0, t, t}, {t, t, t}, {0, 0, t}}]];
setup3 = strip[step];
line = 
        Graphics3D[{Goldenrod, Thickness[ .018], 
            Line[{{0, (step/2), step}, {(step/2), (step/2), 
                  step}}]}];
point1 = 
        Graphics3D[{Red, 
            Text["{0,y,z}", {0, (step/2), step +  .12}]}];
point2 = 
        Graphics3D[{Red, 
            Text["{y,y,z}", {step -  .12, (step/2) -  .10, 
                step +  .2}]}];
point3 = 
        Graphics3D[{Red, 
            Text["{0,z,z}", { .2, step, step +  .10}]}];
point4 = 
        Graphics3D[{Red, 
            Text["{0,0,z}", { .16, 0, step +  .14}]}];
Show[setupyz, setup1, setup2, setup3, line, point1, point2, point3, 
        point4, Axes -> True, Boxed -> False, 
        AxesLabel -> {"x", "y", "z"}, 
        Boxed -> False];

Ç åãêÜñóéá óôï Üîïíá 0z ôïìÞ (ôï ìðëÝ ôñßãùíï) "óáñþíåé" ôï óôåñåü ìáò ùò åîÞò:

Clear[strip, setup3, step]
step =  .1
strip[t_] := 
        Graphics3D[
          Polygon[{{0, 0, t}, {0, t, t}, {t, t, t}, {0, 0, t}}]];
setup3 = Table[strip[t], {t, step, 1, step}];
Show[setupyz, setup1, setup2, setup3, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}, 
        Boxed -> False];

Clear[strip, setup3, step]
step =  .05
strip[t_] := 
        Graphics3D[
          Polygon[{{0, 0, t}, {0, t, t}, {t, t, t}, {0, 0, t}}]];
setup3 = Table[strip[t], {t, step, 1, step}];
\(\(method1 = 
	
        Show[setupyz, setup1, setup2, setup3, Axes -> True, 
          AxesLabel -> {"x", "y", "z"}, 
          Boxed -> False];\)\)

Ôï ßäéï "óÜñùìá" ôïõ ôåôñáÝäñïõ ìðïñïýìå íá ôï åðéôý÷ïõìå üðùò öáßíåôáé óôï ðáñáêÜôù ó÷Þìá

Clear[strip2, setup4, strip2, step]
step =  .5;
strip2[t_] := 
        Graphics3D[
          Polygon[{{0, t, t}, {0, t, 1}, {t, t, 1}, {t, t, t}, {0, t, 
                t}}]];
setup4 = Table[strip2[t], {t, step, 1, step}];
line = 
        Graphics3D[{Goldenrod, Thickness[ .01], 
            Line[{{0, step, step +  .25}, {step, step, 
                  step +  .25}}]}];
point1 = 
        Graphics3D[{Red, 
            Text["{0,y,z}", {0, step -  .15, step +  .30}]}];
point2 = 
        Graphics3D[{Red, 
            Text["{y,y,z}", {step +  .20, step -  .15, 
                step +  .35}]}];
point3 = 
        Graphics3D[{Red, 
            Text["{0,y,y}", {0, step -  .15, step +  .10}]}];
point4 = 
        Graphics3D[{Red, 
            Text["{0,y,1}", {0, step -  .15, 1.10}]}];
Show[setupyz, setup1, setup2, setup4, line, point1, point2, point3, 
        point4, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}, 
        Boxed -> False];

Ôþñá, ç åãêÜñóéá óôïí Üîïíá 0y ôïìÞ åßíáé Ýíá ïñèïãþíéï ðáñáëëçëüãñáììï.

Clear[strip2, setup4, strip2, step]
step =  .1;
strip2[t_] := 
        Graphics3D[
          Polygon[{{0, t, t}, {0, t, 1}, {t, t, 1}, {t, t, t}, {0, t, 
                t}}]];
setup4 = Table[strip2[t], {t, step, 1, step}];
Show[setupyz, setup1, setup2, setup4, Axes -> True, 
        AxesLabel -> {"x", "y", "z"}, 
        Boxed -> False];

Clear[strip2, setup4, strip2, step]
step =  .05;
strip2[t_] := 
        Graphics3D[
          Polygon[{{0, t, t}, {0, t, 1}, {t, t, 1}, {t, t, t}, {0, t, 
                t}}]];
setup4 = Table[strip2[t], {t, step, 1, step}];
method2 = 
        Show[setupyz, setup1, setup2, setup4, Axes -> True, 
          AxesLabel -> {"x", "y", "z"}, 
          Boxed -> False];

Óôï äåýôåñï "óÜñùìá" ïëïêëçñþóáìå ðñþôá ùò ðñïò x (êßôñéíç ãñáììÞ). Óôï äåýôåñï âÞìá êñáôÜìå ôï y óôáèåñü êáé ïëïêëçñþíïõìå ùò ðñïò z  ìå üñéá ôï y êáé ôï 1. ÔÝëïò ïëïêëçñþíïõìå ùò ðñïò y áðü ôï 0 Ýùò ôï 1. 
ÔåëéêÜ, åÜí áëëÜîïõìå ôçí óåéñÜ ïëïêëÞñùóçò áðü 
åßíáé dx dy dz óå dx dz dy ôï ïëïêëÞñùìá 

\(∫\_0\%1\(∫\_0\%z\(∫\_0\%y\ \ xy\^2\ z\^3\)\)\) dx dy dz  ãßíåôáé    (∫_0^1)(∫_y^1)(∫_0^y)x (y^2) (z^3)dx dz dy. 
ÃéÜ ôïõ ëüãïõ ôï áëçèÝò :

Clear[x, y, z]
(∫_0^1)(∫_y^1)(∫_0^y) 
                x (y^2) (z^3) \[DifferentialD]x \[DifferentialD]z \[DifferentialD]y  ==  (∫_0^1)(∫_0^z)(∫_0^y) 
                x (y^2) (z^3) \[DifferentialD]x \[DifferentialD]y \[DifferentialD]z

◆ÊÝíôñï ìÜæáò êáé ñïðÞ áäñáíåßáò

ÅÜí D åßíáé Ýíá 3-äéÜóôáôï óôåñåü êáé  δ(x,y,z) ìßá óõíå÷Þò óõíÜñôçóç ðïõ äßíåé ôçí ðõêíüôçôá êáé åßíáé ïñéóìÝíç ãéÜ êÜèå óçìåßï (x,y,z) óôï D, ôüôå Ý÷ïõìå ôïõò ðáñáêÜôù ôýðïõò:

ÌÜæá:		M = \(TraditionalForm\`∫\(∫∫\_\(\(\ \)\(D\)\)\)\)  δ(x,y,z) dV
   
   
ÊÝíôñï ìÜæáò:
	
          x*=((∫∫(∫_( D))x  δ(x, y, z)dV)/(\(\(M\)\(\ \)\(
\)\)))
          y*=((∫∫(∫_( D))y δ(x, y, z)dV    )/(M   )) 
      
          z* = ((∫∫(∫_( D)) z  δ(x, y, z)dV    )/(M   ))
	
	
ÑïðÞ áäñáíåßáò ùò ðñïò ôïõò Üîïíåò :

(Subscript[I, x])=∫∫(∫_( D))  ((y^2) + (z^2)) δ(x,y,z) dV

(Subscript[I, y])=∫∫(∫_( D))  ((x^2) + (z^2)) δ(x,y,z) dV

(Subscript[I, z])=∫∫(∫_( D))  ((x^2) + (y^2)) δ(x,y,z) dV	

	
ÑïðÞ áäñáíåßáò ùò ðñïò ôá åðßðåäá :

(Subscript[I, xy])=∫∫(∫_( D))  (z^2) δ(x,y,z) dV


(Subscript[I, yz])=∫∫(∫_( D))  (x^2) δ(x,y,z) dV


(Subscript[I, xz])=∫∫(∫_( D))  (y^2) δ(x,y,z) dV

ÐáñÜäåéãìá 1

Õðïëïãßóôå ôéò óõíôåôáãìÝíåò ôïõ êÝíôñïõ ìÜæáò ôïõ óôåñåïý R ðïõ öñÜóóåôáé áðü ôéò åðéöÜíåéåò 
          y= ((x^2)+(z^2)) êáé y = 18 - ((x^2)+(z^2)), 
åÜí ç ðõêíüôçôá óå êÜèå óçìåßï äßíåôáé áðü ôçí óõíÜñôçóç  δ(x,y,z) = (x^2) + (y^2)+ (z^2)

Tï óôåñåü R åßíáé ôï ßäéï óôåñåü ìå áõôü ðïõ ÷ñçóéìïðïéÞóáìå óôá ðáñáðÜíù ðáñáäåßãìáôá ïëïêëÞñùóçò.

Clear[δ, x, y, z, g1, 
      g2]
g1[x_, z_] = (x^2) + (z^2);
g2[x_, z_] = 
        18 - ((x^2) + (z^2));
δ[x_, y_, z_] = 
        (x^2) + (y^2) + (z^2);
ÌÜæá=(∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z]))δ[x, y, 
                z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x

xÊÝíôñïõÌÜæáò=(((∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z]))x δ[
                  x, y, z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x)/(ÌÜæá))

yÊÝíôñïõÌÜæáò=(((∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z])) y δ[x, y, 
                  z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x)/(ÌÜæá))

zÊÝíôñïõÌÜæáò=(((∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z])) z δ[x, y, 
                  z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x)/(ÌÜæá))

ÐáñÜäåéãìá 2

Õðïëïãßóôå ôçí ñïðÞ áäñáíåßáò ôïõ óôåñåïý R ãýñù áðü ôïí Üîïíá 0y ðïõ öñÜóóåôáé áðü ôéò åðéöÜíåéåò 
          y= ((x^2)+(z^2)) êáé y = 18 - ((x^2)+(z^2)), 
åÜí ç ðõêíüôçôá óå êÜèå óçìåßï äßíåôáé áðü ôçí óõíÜñôçóç  δ(x,y,z) = (x^2) + (y^2)+ (z^2)

Clear[δ, x, y, z, g1, 
      g2]
g1[x_, z_] = (x^2) + (z^2);
g2[x_, z_] = 
        18 - ((x^2) + (z^2));
δ[x_, y_, z_] = 
        (x^2) + (y^2) + (z^2);
yÑïðÞÁäñáíåßáò=(∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z]))((x^2) + (z^2)) δ[x, y, 
                z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x

ÐáñÜäåéãìá 3

Õðïëïãßóôå ôçí ñïðÞ áäñáíåßáò ôïõ óôåñåïý R, ãýñù áðü ôï åðßðåäï xy, ðïõ öñÜóóåôáé áðü ôéò åðéöÜíåéåò 
          y= ((x^2)+(z^2)) êáé y = 18 - ((x^2)+(z^2)), 
åÜí ç ðõêíüôçôá óå êÜèå óçìåßï äßíåôáé áðü ôçí óõíÜñôçóç  δ(x,y,z) = (x^2) + (y^2)+ (z^2)

Clear[δ, x, y, z, g1, g2]
g1[x_, z_] = (x^2) + (z^2);
g2[x_, z_] = 18 - ((x^2) + (z^2));
δ[x_, y_, z_] = (x^2) + (y^2) + (z^2);
xyÑïðÞÁäñáíåßáò = (∫_(-3)^3)(∫_(-(Sqrt[9 - (x^2)]))^((Sqrt[9 - (x^2)])))(∫_(g1[x, z])^(g2[x, z]))(z^2) δ[x, y, 
                z] \[DifferentialD]y \[DifferentialD]z \[DifferentialD]x