NAME

       integrate - integrate a function or an expression

DESCRIPTION

       Integrate  an  expression  over  a domain by using a quadrature formulae.  There are three
       main usages of the integrate function, depending upon the type  of  the  expression.   (i)
       When  the  expression  is  a  numerical one, it leads to a numerical value.  (ii) When the
       expression involves a symbolic test-function see test(2), the result  is  a  linear  form,
       represented  by the field class.  (iii) When the expression involves both  symbolic trial-
       and test-functions see test(2), the result is a bilinear form, represented  by  the  field
       class.

SYNOPSYS

        template <class T, class M, class Expr>
        T integrate (const geo_basic<T,M>& omega, Expr expr,
          quadrature_option_type qopt)

        template <class T, class M, class Expr>
        field integrate (const geo_basic<T,M>& omega, VFExpr expr,
          quadrature_option_type qopt)

        template <class T, class M, class Expr>
        form integrate (const geo_basic<T,M>& omega, VFExpr expr,
          form_option_type fopt)

EXAMPLE

         Float f (const point& x);
         ...
         quadrature_option_type qopt;
         Float value = integrate (omega, f, qopt);
         field lh = integrate (omega, f*v, qopt);

       The  last  argument  specifies  the  quadrature  formulae  used for the computation of the
       integral.  The expression can be any function, classs-function or any linear or  nonlinear
       field expression see field(2).

DEFAULT ARGUMENTS

       In  the  case  of  a linear form, the domain is optional: by default it is the full domain
       definition of the test function.

         field l1h = integrate (f*v, qopt);

       When the integration is perfomed on a subdomain, this subdomain simply replace  the  first
       argument and a domain name could also be used:

         field l2h = integrate (omega["boundary"], f*v, qopt);
         field l3h = integrate ("boundary", f*v, qopt);

       The  quadrature formulae is required, except when a test and/or trial function is provided
       in the expression to integrate.  In that case, the quadrature formulae is deduced from the
       space  containing the test (or trial) function.  When a test function is suppied, let k be
       its polynomial degree.  Then the default quadrature is choosen to be exact  at  least  for
       2*k+1  polynoms.  When both a test and trial functions are suppied, let k1 and k2 be their
       polynomial degrees.  Then the default quadrature is choosen  to  be  exact  at  least  for
       k1+k2+1  polynoms.   Also,  when  the expression is a constant, the quadrature function is
       optional: in that case, the constant is also optional and the following call:

         Float meas = integrate (omega);

       is valid and returns the measure of the domain.

SEE ALSO

       test(2), test(2), field(2)