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When you input f(x,a) into numerical_integral, it already knows that a is a parameter. The optional input params is a way for you to give the value of the parameters.

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

I'm not sure exactly what happens when you evaluate

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

--- it would seem that the parameter disappears (or is given the value 1). This means your g(x,a) defined as

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

doesn't depend on a, so g(x,anything) will always get you the same value.

For some reason, it doesn't this way:

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

There's still something to figure out here!

A few observations.

A note. When you feed numerical_integral with the input f(x,a) into numerical_integral, it ~~already~~ detects that the input has two variables and knows ~~that a~~ one of them is a ~~parameter.~~ parameter. The optional input params ~~is a way for you to~~ lets one give the ~~value~~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)

I'm 3.330669073875469e-14)
What is not ~~sure exactly what happens when~~ 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?

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)
 --- Not sure if this behaviour is wanted or a bug.
  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 would seem that the parameter disappears (or is given the value 1).
This means your g(x,a) defined as
 sage: g(x,a)=numerical_integral(f(x,a),0,1, params=[a])
 doesn't now looks like x was set
to a before integrating with respect to a.
  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(x,anything)g(anything,anything) will always get you always
return the same value.
  For some reason, it the "same" g defined using def doesn't this way:
 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)
 There's still something to figure out here!
 3.330669073875469e-14)

          3    No.3 Revision 
      
    
        
        updated 2013-05-01 07:21:25 +0200  
 
 
  
 
 
 
  A few observations.
  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)3.330669073875469e-14)
 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?
 A surprise. When you evaluate
 
sage: numerical_integral(f(x,a),0,1, params=[a])
(0.3333333333333333, 3.700743415417188e-15)
 
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)
 Not sure if this behaviour is wanted or a bug.
  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)
 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.
  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])
 params=[a])
 this does not depend on x or a, so g(anything,anything) will always
return the same value.
  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)3.330669073875469e-14)
 
 
 
          4    No.4 Revision 
      
    
        
        updated 2013-05-01 07:33:18 +0200  
 
 
  
 
 
 
  A few observations.
  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)
 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?
 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)
 Not sure if this behaviour is wanted or a bug.
 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.
 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.
 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)
 
 
 
          5    No.5 Revision 
      
    
        
        updated 2013-05-01 07:50:01 +0200  
 
 
  
 
 
 
  A few observations.observations can help explain what you are experiencing.
  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)
 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?
 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)
 Not sure if this behaviour is wanted or a bug.
 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.
 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.
 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)
 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=[1])
(0.3333333333333333, 3.700743415417188e-15)
 and
 sage: f(x,a)=a*x^2
sage: numerical_integral(f(x,a),0,1, params=[1])
(0.5, 5.551115123125783e-15)
 Can you change which variable is the main variable and which are parameters?
 I haven't figured that out yet.