Average Energy density and the Poynting vector of an EM wave

[edfink1]

Homework Statement

Average Energy density and the Poynting vector of an EM wave

Relevant Equations

energy density of the EM fields, Poynting vector

Hi,
In Problem 9.12 of Griffiths Introduction to Electrodynamics, 4th edition (Problem 9.11 3rd edition), in the problem, he says that one can calculate the average energy density and Poynting vector as

using the formula

I don't really understand how to do this. I show my attempt below, but I don't think it is good. Can someone explain if it is really all there is to it?

 

[Nate810]

My attempt: To calculate the average energy density, we use the formulaU = 1/2 * ɛ₀* E² + 1/2 * μ₀ * B². Where E and B are the electric and magnetic fields, respectively. Substituting the fields for this problem gives usU = 1/2 * ɛ₀ * (E₁ + E₂)² + 1/2 * μ₀ * (B₁ + B₂)².To calculate the Poynting vector, we use the formulaS = ɛ₀*E×BSubstituting the fields for this problem gives usS = ɛ₀ * (E₁ + E₂) × (B₁ + B₂).