Finding spectral exitance over a finite wavelength band using Planck's law

[cant_stop_shaking]

Hello,

My question is in regards to Planck's Law and a blackbody:

For the single lambda case I can readily find the spectral exitance. Alternately, if I substitute to create an integral in the form of x^3 / (e^x - 1) and integrate over all lambda, I reach Stefan-Boltzmann. No problems there, but I am really interested in finding the (average?) spectral exitance over a small, finite spectral band. Any tips on how to go about this?

Thanks in advance.

 

[vincentchan]

i am not quite sure what your question is, but it seems to me you are looking for the answer of this integral:
$$\int_{b}^a \frac{x^3}{e^x-1} dx$$,
the above integral has no exact solution... unless a=0, b=infinity, or a=b... the best you could do is use numerical analysis...

 

[dextercioby]

If you don't have a computer software to give you the result,learn that Debye integral ($D_{3}$) values are tabulated...

Daniel.

 

[cant_stop_shaking]

Thank you both for your quick responses. To state my question more directly: "What is the area under the curve generated by a blackbody at some given temperature for some finite lambda range?" (Say in the visible region only.)

I can generate an approximate answer by using small increments of area and summing, but I wanted to compare my answer the solution obtained from a definite integral. However, from your answers I believe it is not possible (I do not have any computer software - just pencil and paper!).

I have taken your advice and will study more about numerical analysis and Debye integrals. Thank you again for your help.