This paper presents an explicit integration scheme to compute  the stiffness matrix of an eight node linear convex quadrilateral element  for plane problems using symbolic mathematics and an automatic generation of all quadrilateral mesh technique , In finite element analysis, the boundary problems governed by  second order linear partial  differential equations,the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the  matrix  of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in bivariates  over an eight node  2-square (-1 ).In this paper,we have computed these integrals  in exact and digital forms  using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme is illustrated by computing the Prandtl stress function values and the torisonal constant for the square cross section by using the eight node linear convex quadrilateral finite elements.An automatic  all quadrilateral  mesh generation  techniques  for the eight node linear convex quadrilaterals is  also developed for this purpose.We have presented a complete program which automatically discritises the arbitrary triangular domain into all eight node linear convex quadrilaterals  and applies the so generated nodal coordinate and element connection data to the above mentioned torsion problem.