e^π > π^e follows from bounding the integral of 1/x over [e, π].
The integral equals ln(π) - 1 and is less than (π - e)/e, the rectangle area at maximum height 1/e.
Simplification gives ln(π) < π/e, then e ln(π) < π or ln(π^e) < π. Exponentiation yields π^e < e^π.
This bounding technique supports inequality proofs for exponential expressions in physics modeling of variable rates and in economics for growth comparisons.