# Definite integral depends on the variable of integration

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### Andrey Novoseltsev

Feb 3, 2012, 5:10:07 PM2/3/12
to sage-devel
Hello,

It is a known bug? Or maybe not considered a bug at all? Quick search
does not show anything related...

sage: f(x) = x
sage: f
x |--> x
sage: integral(f, x)
x |--> 1/2*x^2
sage: integral(f, x, 0, 1)
x |--> 1/2

The last line shows 1/2 as a function of x, but it is not a function
anymore, it is a number. It gets worse with multivariate functions:

sage: f(x, y) = x + y
sage: f
(x, y) |--> x + y
sage: integral(f, x, 0, 1)
(x, y) |--> y + 1/2
sage: _(3)
y + 1/2

I think that here integral definitely should return a function of y
only and if I evaluate it at 3 I get 3.5.

Best regards,
Andrey

### kcrisman

Feb 3, 2012, 6:04:32 PM2/3/12
to sage-devel

On Feb 3, 5:10 pm, Andrey Novoseltsev <novos...@gmail.com> wrote:
> Hello,
>
> It is a known bug? Or maybe not considered a bug at all? Quick search
> does not show anything related...
>
> sage: f(x) = x
> sage: f
> x |--> x
> sage: integral(f, x)
> x |--> 1/2*x^2
> sage: integral(f, x, 0, 1)
> x |--> 1/2
>
> The last line shows 1/2 as a function of x, but it is not a function
> anymore,

I'll let others comment on this design decision (which does make sense
in terms of the logic of callable versus not-callable symbolic
expressions).

>
> sage: f(x, y) = x + y
> sage: f
> (x, y) |--> x + y
> sage: integral(f, x, 0, 1)
> (x, y) |--> y + 1/2
> sage: _(3)
> y + 1/2
>
> I think that here integral definitely should return a function of y
> only and if I evaluate it at 3 I get 3.5.

This might be a different problem. Shouldn't this give a syntax
error, folks?

sage: h(x,y) = x+y
sage: h(3)
y + 3

??? I thought the whole *point* of the h(x,y) notation was to specify
the order - but then you'd figure it would also specify the number of
entries. I guess this is a shorthand, perhaps, but then I don't know
how to get just h(,3) (which is naturally an error), which you'd want
by symmetry. I guess h(x,3) works, but then we should (perhaps?)
require h(3,y) as well?

- kcrisman

### P Purkayastha

Feb 4, 2012, 6:31:59 AM2/4/12
I believe this is more or less consistent behavior throughout Sage. For example:

sage: derivative(x, x)
1
sage: type(derivative(x, x))
<type 'sage.symbolic.expression.Expression'>

### P Purkayastha

Feb 4, 2012, 6:36:40 AM2/4/12

On Saturday, February 4, 2012 7:04:32 AM UTC+8, kcrisman wrote:

This might be a different problem.  Shouldn't this give a syntax
error, folks?

sage: h(x,y) = x+y
sage: h(3)
y + 3

???  I thought the whole *point* of the h(x,y) notation was to specify
the order - but then you'd figure it would also specify the number of
entries.  I guess this is a shorthand, perhaps, but then I don't know
how to get just h(,3) (which is naturally an error), which you'd want
by symmetry.  I guess h(x,3) works, but then we should (perhaps?)
require h(3,y) as well?

- kcrisman

I think h(3) should give syntax error. It doesn't make sense mathematically, and otherwise, it is ambiguous.

### Andrey Novoseltsev

Feb 4, 2012, 10:43:44 AM2/4/12
to sage-devel
Derivative of a function depending on x is another function depending
on x. If it happens to be a constant or some expression with
parameters not depending on x explicitly - it is a coincidence. Same
for indefinite integrals.

But in definite integrals the variable of integration is a dummy one
and does not exist out of the integral!

### Keshav Kini

Feb 4, 2012, 12:24:00 PM2/4/12
On Saturday, February 4, 2012 11:43:44 PM UTC+8, Andrey Novoseltsev wrote:
But in definite integrals the variable of integration is a dummy one
and does not exist out of the integral!

+1. This behavior makes no sense to me.

-Keshav

----

### Jason Grout

Feb 4, 2012, 1:21:57 PM2/4/12
On 2/4/12 11:24 AM, Keshav Kini wrote:
> On Saturday, February 4, 2012 11:43:44 PM UTC+8, Andrey Novoseltsev wrote:
>
> But in definite integrals the variable of integration is a dummy one
> and does not exist out of the integral!
>
>
> +1. This behavior makes no sense to me.

So what happens in a case like this?

sage: f(x,y,z)=sin(x*y*z)

sage: integrate(f,(y,0,3))

Does the result become a function

(x,z) |--> whatever?

I can see it making sense, but I can also see it causing problems.

Thanks,

Jason

### Andrey Novoseltsev

Feb 4, 2012, 1:26:04 PM2/4/12
to sage-devel
What kind of problems can it cause?.. We have a function with
variables which are named and have a particular order. If one of them
is gone, the rest keeps both their names and the same ordering.

### Jason Grout

Feb 4, 2012, 1:30:12 PM2/4/12

I was thinking of confusion for the user, because the user has to be
more careful about how they call the function because the arguments
change. Now, this can be alleviated by the user using keyword
arguments, or paying attention and being aware of integration removing a
variable. Again, I don't think this is insurmountable, and arguably
it's desirable to make the user think about how the function domain is
changing, but I feel obligated to bring up the issue as a point against
changing things in a backwards-incompatible way.

Thanks,

Jason

### Andrey Novoseltsev

Feb 4, 2012, 1:50:08 PM2/4/12
to sage-devel
On Feb 4, 11:30 am, Jason Grout <jason-s...@creativetrax.com> wrote:

> I was thinking of confusion for the user, because the user has to be
> more careful about how they call the function because the arguments
> change.  Now, this can be alleviated by the user using keyword
> arguments, or paying attention and being aware of integration removing a
> variable.  Again, I don't think this is insurmountable, and arguably
> it's desirable to make the user think about how the function domain is
> changing, but I feel obligated to bring up the issue as a point against
> changing things in a backwards-incompatible way.
>
> Thanks,
>
> Jason

I think it is much more confusing to supply an extra argument (whose
value does not matter at all!) then to think about change in the
domain. From a mathematical viewpoint it seems unambiguous to me that
the current behaviour is wrong and should be changed, and so far there
were no implementation reasons for keeping it. If dropping a variable
and allowing unnamed evaluation is confusing, it may not be allowed as
well, but I personally think that it is quite desirable.

I hit it while preparing for my multivariate calculus class where I
teach students that taking a function f(x,y) over some region R and
integrating it in y from g(x) to h(x) gives the area of the cross-
section A(x). If we integrate it in x from u(y) to v(y), we get a
perpendicular cross-section B(y). If we now integrate one more time,
we get a number V - the volume under the graph. Sage output suggest
that we are still getting functions of (x,y) - still over R or what is
its new domain?.. Why would any user try to evaluate, say, V(2,1) and
what would be the expected result? Would this user expect V(3,42)
return the same value? If yes, how about just using V with no
arguments? If no - the user does not seem to understand what is going
on. Raising an error for an attempt to evaluate V(2,1) can be a very
useful cue in this case.

So - I propose opening a ticket to fix this.

Thank you,
Andrey

### Jason Grout

Feb 4, 2012, 1:53:21 PM2/4/12

I think you have a pretty convincing argument for changing this. +1 to
opening the ticket and changing it.

Jason