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)