COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Radioactive decay
The mean lifetime uses improper integral.
one unit for u = 100 user units
one unit for v = 172 user units
The origin of the figure is at (0,0)
©
Samuel Dagan
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u=t/τ
v=N/N0
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v=N/N0
u=t/τ
0∫1dv=1
<u>=1
Radioactive decay
In the expression of a radioactive
decay (N/N0) = exp(−t/τ) , t
is time and τ
is a constant
of the decay with the dimensions of time, N is the
number of
the remaining atoms and N0 is the original number of atoms.
By using the
variables u=t/τ
and v=N/N0 , the above expression becomes
v(u)=exp(−u).
Its inverse is u(v)=−ln(v) . It will be plotted
next with the appropriate scales.
The reflecting mirror is also
shown. <u> will be
obtained next.
By definition
<u> = (0∫1udv) / (0∫1dv) = 1. From u = t/τ , it follows
that <t> = τ.
Next we return to v(u)=exp(−u)
and u(v)=−ln(v) for
further comparisons.
We'll see next the graphical
presentation showing that 0∫1udv =
0∫∞vdu .
Indeed the corresponding areas under
the functions are the same and
can be interchanged by mirror
reflection exactly as the functions.
Is it a general rule that the definite
integrals of a function and its inverse
are equal? The answer is no, and
one has to inspect each case.
This will be illustrated next
by translating v=v(u) to the
right,
accompagnied by the
appropriate translation of the inverse.
Although the new u(v) is
the inverse of the new v(u) ,
the
integral under the
new u(v) is
larger.
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