160        Triple Integration & its Applications                                          Chapter -6
                                                                              6
                Triple Integration & its Applications
               1.        TRIPLE INTEGRATION
                         In the preceeding sections we defined and discussed the properties of single and double integrals for functions
                         of single variable and two variables respectively. In this section, we will define triple integrals for functions of
                         three variables.
               •         The triple integral of  f (x, y,z)  is denoted by          f (x, y,z)dV  and is defined as
                                                                               
                                                                                D
                                                    g2(x, y)
                              f (x, y,z)dV               f (x, y,z)dzdA where D is a simply xy-solid with upper surface z  g (x, y)
                                                                                                                                          2
                                                 
                          D                     R g (x, y)              
                                                    1                   
                         and lower surface z  g (x, y), and let R be the projection of D on the xy-plane.
                                                     1
               •         If D is a rectangular box defined by the inequalities a  x  b, c  y  d, k  z  l , then
                                                b d l
                              f (x, y,z)dV          f (x, y, z)dzdydx
                                             
                          D                     a c k
                         Volume element
                            dV  dzdydx  dzdxdy  dxdydz  dxdzdy                                                        z
                         
                                                                                                                  z
                               dydxdz  dydzdx                                                                     k            y
                                                                                   Cuboidal                           x
                            V (x)(y)(z)                                     sub-region
                                                                                     V                      x
                                                                                         k                       k
                                                                                                   yk
                         Evaluating Triple Integrals :
               Step 1.Find an equation z  g (x, y) for the upper surface and an equation z  g (x, y) for the lower surface of G.
                                                   2                                                          1
                         The functions g (x, y) and g (x, y)determine the lower and upper z-limits of integration.
                                           1               2
               Step 2.Make a two-dimensional sketch of the projection R of the solid on the xy-plane. From this sketch determine
                         the limits of integration for the double integral over R.
               Step 3.After finding limits of integration, do the integration as in case of double integral i.e., partial integration.
                         Geometric Interpretation:
                         From the definition of the triple integral
                                 dV  volume of the region D.
                             
                              D
                                                        Chapter-6                                                Triple  Integration & its Applications                              161
                                                 1 2 3
                   Ex.1: Evaluate  :                    xyzdzdxdy
                                                 
                                                 0 0 0
                   Soln.       Here limit of z : 0  z  3
                               Here limit of x : 0  x  2
                               Here limit of y : 0  y 1
                                      1   2   3                           1   2                      1
                                                                                      
                                             zdzxdx ydy                      9xdx ydy  36 ydy  36
                                                                                    
                                                                       2                         4               8
                                      0 0  0                          0  0                     0
                                                            
                                            1 1 2
                   Ex.2: Evalute                   x2z2ydzdxdy
                                             
                                           1 0 1
                                1 1 2                          1       1         2                 1 7
                   Soln.               x2z2ydzdxdy              ydy x2dx z2dz 0  0
                                                                                             3 3
                                1 0 1                        1       0         1
                                           1/2  1
                   Ex.3: Evalute                   zxsinxydzdydx
                                             
                                           1/3 0 0
                             1/2  1                            1/2   1
                                     zxsinxydzdydx                   zdzxsin(xy)dydx
                   Soln.                                                      
                                                               
                             1/3 0 0                            1/3 0  0      
                                                                1/2                                 1/2  
                                                                     1xsin(xy)dydx  1                  sin(xy)dyxdx
                                                                                                                           
                                                                  2                             2  
                                                                1/3 0                                1/3 0                 
                                                               11/2 cosxy                      11/2                               1         sinx 1/2
                                                                                   xdx             (1cosx)dxss x                               
                                                                                                    
                                                               2           x                   2                                  2             
                                                                                   0                                                                 
                                                                  1/3                               1/3                                                  1/3
                                                                                                 
                                                                          sin              sin      
                                                                11             2     1           3          1 1        1      1           1 1        1         1 1 1
                                                                                                                                                                         
                                                                                  
                                                                                                                                                                
                                                                2 2                  3                      26 2                      26 2 43 
                                                                                                                                                                       
                                                                                                    
                                           /4 1 x2
                   Ex.4: Evalute                    xcosydzdxdy
                                              
                                            0 0 0
                                /4 1 x2                         /4 1                             /4 1
                                         xcosydzdxdy                 [z]x2xcos ydxdy                   x3cosydxdy
                   Soln.                                                   0                         
                                                               
                                 0 0 0                            0 0                               0 0
                                                                /4            1              1       1       1
                                                                   cosydy x3dx
                                                                                                   
                                                                                              2     4     4 2
                                                                 0             0
                   Ex.5: Evaluate                  ydV,where G  is the solid enclosed by the plane z = y, the xy-plane, and the parabolic cylinder y
                                             
                                               G
                                          2 
                               = 1 – x .
                   Soln.       Limit of z, 0  z  y
                               Limit of y, 0  y 1 x2
                   162          Triple Integration & its Applications                                                        Chapter -6
                             Limit of x, 1 x 1                                                                                            y
                                               y
                                   ydv          ydzdxdy
                                       
                               G            R 0                                                                                                 R
                                           1 1x2               1 1                                                                                          x
                                                 y2dydx            (1 x2)3dx
                                                              3
                                           1 0                    1
                                           2 1                                     2        1    3           32
                                               (1 x6 3x4 3x2)dx                 1  1
                                                                                                       
                                           3                                      3        7    5           70
                                              1                                                       
                                                                          9 3 x z
                  Ex.6: Change the order of the integral                            f (x, y,z)dydzdx
                                                                           
                                                                          0  0   0
                  Soln.      Limit of x, 0  y  z                                                                                 z
                             Limit of y, 0  z  3           x                                                                    3
                             Limit of z, 0  x  9
                             i.e., Region bounded by y = z, z  3                  x,                                             0                  9      x
                             xz-plane, yz-plane
                                   9 3 x z                            3 (3z)2 z
                                             f (x, y,z)dydzdx                    f (x, y,z)dydzdx                                 y
                                                                    
                                   0  0   0                            0   0   0                                                   3      y 3 x
                                                                       9 3 x 3 x
                                                                                    f (x, y,z)dzdydx
                                                                              
                                                                       0   0     y                                                 0                  9      x
                                                                        3 (y3)2 3 x
                                                                                      f (x, y,z)dzdydx
                                                                                 
                                                                        0    0     y
                     •       To change the order of integral First we consider
                             (i)  region of integral
                             (ii) Go parallel to axis suppose we first consider limit of z then we go parallel to z-axis. Put lower limit entering
                             first surface in x, y relation and put upper limit ending with surface in x, y relation.
                             (iii) Then we find out the relation in x and y with the help of z
                             (iv) Now we work just as double integral
                  1.1        Cylindrical coordinates
                             Cylindrical coordinates of a point P is (r,, z)  as shown in the figure. If cartesian coordinates of P be
                             (x, y, z)  then
                                                                                                                                  z
                              x  rcos, y  rsin,z  z                                                                                           P(r, , z)
                           r2  x2  y2 and tan  y / x                                                                                           z
                                                                                                                              O                                 y
                             For one-to-one correspondence from rectangular to                                              x             r
                       
                                                                                                                                                       x
                             cylindrical coordinates we take r 0 and 02.                                                       y             P
                                                                                                                  x
                                             Chapter-6                                   Triple  Integration & its Applications                163
                        Although, there can be six order of integration, similar to the rectangular coordinates. We take only
                    
                         rdzdrd as it gives easier integration.
                        Simple geometries in cylindrical coordinates:                                                    z
               1.        r  r  (a positive constant)
                              0
                        geometry: a right circular cylinder with z-axis its axis and radius as r .                                  y
                                                                                                          0                    r=r
                                                                                                                  x                0
                                                                                                           z
               2.         (0 2) (a constant)
                               0       0
                        geometry : a plane containing z-axis
                        and making an angle 0  with positive x-axis.                                  O                y
                                                                                                          
                                                                                                           0
                                                                                                    x         = 0
               3.        z  z  (a constant)                                                                 z
                                0
                        geometry: a plane perpendicular to the                                              z
                                                                                                             0
                        z-axis and cutting z-axis at (0, 0, z )                                                     y
                                                                  0                                        O
                        Volume element: Cartesian and cylindrical                                     x
                                   Cartesian                                Cylindrical
                                                       z                          rd
                                                              y         z
                         dz                        x                                          dz
                                            dx
                                  dy                                           d       dr
                                                                               r
                         dV dxdydz                                 dV  (rd)(dr)dz  rdzdrd  dzrdrd
                        (Cuboid)                                    (Cylindrical wedge)                         z
                        Integration: For the region, G enclosed by  z  g2(r,)
                                                                                                                      z = g (r, )
                        and  z  g (r,) and having projection R in the xy-plane                                           2
                                     1                                                                          G
                                                          g2(r,)                                                       z = g (r, )
                                                                                                                           1
                                                                g(r,,z)rdz drd
                             g(r,,z)rdzdrd                                   
                                                                                                   
                         
                                                       R  g (r,)                                                      y
                          G                               1                    
                        G : transformed region in              coordinate system.                        x            R
                                                        r  z
                        Transformation to cylindrical coordinates: Transformation from rectangular to cylindrical coordinates.
                              f (x, y, z)dzdxdy         f (r cos,rsin,z)rdzdrd
                                                 
                          D                          G
                        G : transformed region in r,,z coordinates
                                  /2cosr2
               Ex.7: Evalute               rsindzdrd
                                     
                                   0  0  0
                         /2cosr2                   /2cos
                                                               r2
                                  rsindzdrd             [z] rsindrd
               Soln.                                           0
                                                   
                          0  0  0                     0  0