(Solved): Problem 1) The Stefan-Boltzmann law is a statement on the total black-body power \( R \) (energy r ...

Problem 1) The Stefan-Boltzmann law is a statement on the total black-body power \( R \) (energy radiated per second) emitted per unit area. \( R=\frac{c}{4} u(v, T) \), here \( \mathrm{c} \) is the speed of light, which converts energy per unit volume to power per unit area. The law is \( R=\sigma T^{4} \) : In other words, it is the integral of Plank's law \( u(v, T) \) over the frequency \( v \) (or the wavelength \( \lambda, u(\lambda, T) \) ) times \( c / 4 \). Choose the form you like and integrate over appropriate variable ( \( v \) or \( \lambda) \). By change of the integration variable to \( x \) reduce the integral to con?tants times \( \int_{0}^{\infty} \frac{x^{3} d x}{e^{x}-1}=\frac{\pi^{4}}{15} \). Obtain \( \sigma \) and evaluate its numerical value include units.