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MathGroup Archive 2007

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Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77600] Integrate question
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Wed, 13 Jun 2007 07:44:47 -0400 (EDT)

This question aose after replying in the forum for another CAS.
In fact, it is not a question on "how I do something" but rather
asking about some information about "why".

In view of my reply I realized one fundamental difference
between Integrate and the function Int from another CAS.

int( diff(y(x),x)  , x=a..b);

                              b
                             /
                            |   d
                            |   -- y(x) dx
                            |   dx
                           /
                             a

In[11]:=
Integrate[Derivative[1][f][x], {x, a, b}]
Out[11]=
-f[a]+f[b]

I guess all we are familiar with the first fundamental theorem of
calculus.
It holds (http://mathworld.wolfram.com/
FundamentalTheoremsofCalculus.html)
if the function f[x] is continuous on the closed interval [a,b] (or at
least piecewise
continuous in Newton-Leibniz form with limits left and right of the
finite discontinuities).
I am with Mathematica default design. For example

In[12]:=
Integrate[Derivative[1][f][x]*Derivative[2][f][x], {x, a, b}]

Out[12]=
(1/2)*(-Derivative[1][f][a]^2 + Derivative[1][f][b]^2)

In[13]:=
Integrate[Derivative[1][f][x]*g[x] + f[x]*Derivative[1][g][x], {x, a,
b}]

Out[13]=
(-f[a])*g[a] + f[b]*g[b]

In[14]:=
Integrate[Sin[f[x]]*Derivative[1][f][x], {x, a, b}]

Out[14]=
Cos[f[a]] - Cos[f[b]]

I am neither software enginner, nor pure mathematician but this
fundmental difference impressed me a lot! I am familiar
with the issue of generic complex values in mathematica
but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}]
f[x] is continuous in [a,b]?

I will appreciate any comments on this issue.

Greeting from the sunny Athens,

Dimitris



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