On Sat, Dec 6, 2008 at 11:39 AM, Paul Butler <[EMAIL PROTECTED]> wrote:
>
> Currently, taking the integral of a piecewise function in Sage gives
> you the definite integral. I've proposed on trac that the integral of
> piecewise functions be indefinite by default. This would be consistent
> with how integration works on other functions in Sage, as well as
> piecewise functions in Maple and Mathematica.
>
> The main concern is whether the integral of a piecewise function is
> even well-defined. It seems to me that at least for continuous
> piecewise functions, the indefinite integral is well-defined. The
> anti-derivative is well defined, and by the fundamental theorem of
> calculus, the indefinite integral of a continuous function is the
> anti-derivative. As for discontinuous piecewise functions, I'm finding
> it difficult to convince myself either way.
First, the indefinite integral is the anti-derivative, by definition.
What the FTC says is that although the indefinite integral "evalated
at b" is not
well-defined, and the same antiderivative "evalated at a" is also not
well-defined,
their difference *is* well-defined. Moreover, this difference agrees with the
definite integral defined by the Riemann sum between a and b.
Second, I think (but I am not sure), what you want when you say
indefinite intergal is not the indefinite integral but is a function who
derivavtive is the original function, defined as follows:
if the orginial function f(x) is
f1(x), a1<x<=a2,
f2(x), a2<x<=a3,
...
fn(x), an<x<=a{n+1}
(and 0 outside (a1,a{n+1}) then I guess you want to define the integral,
call it F, by
int_{a1}^x f1(t) dt, a1<x<=a2,
int_{a2}^x f2(t) dt, a2<x<=a3,
...
int_{an}^x fn(t) dt, an<x<=a{n+1}
Is this correct? This is not the antiderivative
but it does have the property that F'(x)=f(x).
The antiderivative is only well-defined up to an additive constant.
IMHO, the piecewise defined function of antiderivavtives
int f1(x) dx +C1 , a1<x<=a2,
int f2(x) dx +C2, a2<x<=a3,
...
int fn(x) dx +Cn, an<x<=a{n+1}
does not make sense.
>
> The trac ticket is 4721 ( http://trac.sagemath.org/sage_trac/ticket/4721 )
>
> -- Paul
>
> >
>
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