Help me understand the math behind this

Moneer81

Hi,

while solving for the average energy given by the following formula:

[tex] \overline {E} = \frac {\int_{0}^{\infty} E e^\frac{-E}{kT}dE}{\int_{0}^{\infty} e^\frac{-E}{kT}dE} [/tex]

where E bar is average energy, k is the Boltzmann's constant, and T is temperature

I had to use integration by parts for the numerator.

Integration by parts formula is [tex] \int u dv = uv - \int v du [/tex]

So I made the following choices (and so did my textbook):

[tex] u = E [/tex]

then [tex] du = dE [/tex]

[tex] dv = e^\frac{-E}{kT} [/tex]

and so [tex] v = -kTe^\frac{-E}{kT} [/tex]

Then I proceeded by applying the integration by parts formula, and the integral of the numerator would be:

[tex] \int_{0}^{\infty} E e^\frac{-E}{kT}dE = -EkTe^\frac{-E}{kT} + kT \int_{0}^{\infty} e^\frac{-E}{kT}dE [/tex]

but to my surprise, the book proceeded in the following manner:

[tex] \int_{0}^{\infty} E e^\frac{-E}{kT}dE = kT \left[e^\frac{-E}{kT} \right]
_{0}^{\infty} + kT \int_{0}^{\infty} e^\frac{-E}{kT}dE [/tex]

That first term to the right of the equal sign threw me off....where did it come from?

dextercioby

First

[tex] dv=e^{-\frac{E}{kT}} \ dE [/tex]

then you're right. It has to be some error in the book. Perhaps they meant

[tex] kT \left[e^{-\frac{E}{kT}}\right]_{+\infty}^{0} [/tex]

,that is converting the minus before the whole term into an interchange of integration limits.

Daniel.