# Area under Wien's displacement graph

Why does the area under Wien's displacement graph give Stefan-Boltzmann law for a black body?

I couldn't find any proof of this. (I could just find this expression). I am not aware of the function of Wien's displacement graph as well (I just know that it is between Intensity and wavelength emitted by a black body).

Is there a mathematical way to prove this?

Since the question is a little terse, it is difficult to interpret. I think what must be happening is that the phrase 'Wien's displacement graph' is being used to mean the graph of $$\rho(\omega)$$ as a function of frequency $$\omega$$, where $$\rho(\omega)$$ is the energy density per unit frequency range in thermal or black body radiation. This graph implies Wien's displacement law if one studies it as a function of temperature. And the area under this graph is the total energy density in the radiation, which obeys the Stefan-Boltzmann law, as follows: $$\rho(\omega) = \frac{\hbar}{\pi^2 c^2} \frac{\omega^3}{e^{\beta \hbar \omega}-1},$$ $$u = \int_0^\infty \rho(\omega) d\omega = \frac{k_B^4 T^4}{\pi^2 c^3 \hbar^3} \int_0^\infty \frac{x^3}{e^x - 1} dx = \frac{\pi^2 k_B^4 T^4}{15 c^3 \hbar^3} = \frac{4 \sigma}{c}T^4 .$$ The power per unit area emitted by the surface of a black body is related to this by $$I = \frac{1}{4} u c = \sigma T^4 .$$