symbolic and numeric double integration method

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I'm interpreting your question as: Is there a sage function which will give a symbolic solution to an integral if a closed form exists and a numeric solution otherwise. To my knowledge there is not though the following function may be what you are asking for:

def sym_or_num_Integral(f,variables,lower_bound=[None],upper_bound=[None]):
    if isinstance(variables,list):
        if not isinstance(lower_bound,list):
            lower_bound=[lower_bound]
        while len(lower_bound)!=len(variables):
            if len(lower_bound)>len(variables): break
            lower_bound.append(None)

        if not isinstance(upper_bound,list):
            upper_bound=[upper_bound]
        while len(upper_bound) != len(variables):
            if len(lower_bound)>len(variables): break
            upper_bound.append(None)

        if len(variables)==1:
            return sym_or_num_Integral(f,variables[0],lower_bound[0],upper_bound[0])
        inputs_inner=[variables[0],lower_bound[0],upper_bound[0]]
        inputs_outer=[variables[1:],lower_bound[1:],upper_bound[1:]]
        return sym_or_num_Integral(sym_or_num_Integral(f,*inputs_inner),*inputs_outer)

    result=integral(f,variables,lower_bound,upper_bound)
    if 'integrate' in str(result):
        return result.n()
    else:
        return result

The method of checking if the solution is a closed form is quite hacky and I'd welcome and improvement too it. This function can do any number of integrals so long as you input the parameters as a list, i.e. variables=[x,y], lower_bound=[x_lower,y_lower]. Two examples:

sage: var('x y')
sage: sym_or_num_Integral(sin(x)*y,[x,y],[0,0],[2,2])
-2*cos(2) + 2
sage: sym_or_num_Integral(sin(x*y),[x,y],[0,0],[2,2])
2.1044917239083536

You can have the integral be definite in any subset (must include first variable) of the variable and not the others:

sage: sym_or_num_Integral(sin(x)*y,[x,y],[0,None],[2,None])
-1/2*y^2*(cos(2) - 1)

Was this all you were looking for?