Probability of the Speed of a Molecule in a Group of Traveling Molecules

Question:
In a group of molecules all traveling in the positive z direction, what is the probability that a molecule will be found with a z-component speed between 400 and 401 m/s if m/(2kT ) = 5.62 × 10⁻⁶ s² /m² ? (Hint: You need to find and normalize a one-dimensional distribution function first!)
Solution:
According to the Boltzmann distribution law, the probability of finding a molecule with an energy ε_z in this group of molecules, is proportional to,

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