Numerical integral with multiple parameters

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A few observations can help explain what you are experiencing.

1. A note. When you feed numerical_integral with the input f(x,a), it detects that the input has two variables and knows one of them is a parameter. The optional input params lets one give the values of the parameters.

   sage: var('x a')
   sage: f(x,a)=a*x
   sage: numerical_integral(f(x,a),0,1, params=[1])
   (0.5, 5.551115123125783e-15)
   sage: numerical_integral(f(x,a),0,1, params=[6])
   (2.9999999999999996, 3.330669073875469e-14)
   

2. What is not clear is which of the two variables in f is considered by numerical_integral as being the parameter. Is it x or a? Which is the main variable, the first one or the last one, or can it be specified?

3. A surprise. When you evaluate

   sage: numerical_integral(f(x,a),0,1, params=[a])
   (0.3333333333333333, 3.700743415417188e-15)
   

   it seems that the parameter is transformed into the main variable before integrating: the answer is exactly the integral of $x^2$ (or $a^2$ if the integration is with respect to $a$) from 0 to 1.

   sage: numerical_integral(x^2,0,1,)
   (0.3333333333333333, 3.700743415417188e-15)
   

4. Weirder (if you thought a was the parameter and x the main variable):

   sage: f(x,a)=a*x^2
   sage: numerical_integral(f(x,a),0,1, params=[1])
   (0.5, 5.551115123125783e-15)
   sage: numerical_integral(f(x,a),0,1, params=[a])
   (0.25, 2.7755575615628914e-15)
   

   Now the integral with parameter set to 1 is wrong, unless integration is with respect to $a$ rather than $x$. With params=[a] it now looks like x was set to a before integrating with respect to a.

5. Since numerical_integral(f(x,a),0,1, params=[a]) has a fixed value, when you define

   sage: g(x,a) = numerical_integral(f(x,a),0,1, params=[a])
   

   this does not depend on x or a, so g(anything,anything) will always return the same value.

6. For some reason, the "same" g defined using def doesn't behave the same:

   sage: def g(x,a):
   ....:     return numerical_integral(f(x,a),0,1, params=[a])
   ....: 
   sage: g(x,6)
   (2.9999999999999996, 3.330669073875469e-14)
   

   Here, g(x,6) returns numerical_integral(f(x,6),0,1, params=[6]), in which the function to integrate only has one variable, x, and the optional parameter params=[6] is just ignored.

   You could simply write:

   sage: def g(a):
   ....:     return numerical_integral(f(x,a),0,1)
   ....: 
   sage: g(6)
   (2.9999999999999996, 3.330669073875469e-14)
   

7. So which one is the main variable?

   In numerical_integral(f(x,a),0,1,params=[value]), the function of several variables is treated as a function of one variable with parameters, but the main variable is a, not x.

   You might think the main variable is the last of the arguments of f(x,a) but it seems that it's really the first variable in the lexicographic order.

   Indeed, compare

   sage: f(x,y)=y*x^2
   sage: numerical_integral(f(x,y),0,1, params ...