Asymptotic areas between powers of sine and cosine curves

Motivated by Dombrowski and Dresden's work in 2025, we find the exact values of the limits $$\lim_{k\to\infty}\int_0^{\rho\pi}|\sin^n(kx)-\sin^nx|dx\quad\text{and}\quad \lim_{k\to\infty}\int_0^{\rho\pi}|\cos^n(kx)-\cos^nx|dx$$ for $k,n\in\mathbb{N}$ and $\rho\ge 0$. In addition, we provide several simple recursive formulas which relate these integrals together. The key technique is to locate the zeros of the integrands explicitly, which allows removal of the absolute value and reduces the problem to evaluating limits of telescoping trigonometric sums via asymptotic analysis.