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

        Float integrate (geo domain);
        Float integrate (geo domain, quadrature_option_type qopt);
        Value integrate (geo domain, Expression, quadrature_option_type qopt);

        field integrate (Expression);
        field integrate (Expression, quadrature_option_type qopt);
        field integrate (geo domain, Expression);
        field integrate (geo domain, Expression, quadrature_option_type qopt);

        form integrate (Expression);
        form integrate (Expression, form_option_type qopt);
        form integrate (geo domain, Expression);
        form integrate (geo domain, Expression, form_option_type qopt);

EXAMPLE

       For computing the measure of a domain:

         Float meas_omega = integrate (omega);

       For computing the integral of a function:

         Float f (const point& x);
         ...
         quadrature_option_type qopt;
         qopt.set_order (3);
         Float int_f = integrate (omega, f, qopt);

       The last argument  specifies  the  quadrature  formulae  (see  quadrature_option_type(2))  used  for  the
       computation of the integral.  The function can be replaced by any field-valued expression (see field(2)).
       For computing a right-hand-side of a variational formulation with the previous function f:

         space Xh (omega, "P1");
         test v (Xh);
         field lh = integrate (f*v);

       For computing a bilinear form:

         trial u (Xh);
         test v (Xh);
         form m = integrate (u*v);

       The expression u*v can be replaced by any bilinear expression (see field(2)).

DEFAULT ARGUMENTS

       In  the  case  of  a  linear  or  bilinear form, the domain is optional: by default it is the full domain
       definition of the test function.  Also, the quadrature formulae is optional: by  default,  its  order  is
       2*k+1  where  k  is the polynomial degree of the Xh space associated to the test function v.  When both a
       test u and trial v 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.  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);
         field l3h = integrate ("boundary", f*v);

       For convenience, only the domain name can be supplied.

SEE ALSO

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

rheolef-6.7                                        rheolef-6.7                               integrate(4rheolef)