Recently, I've got interested in limit-integral questions. While playing around with Desmos, I found the formula (Dis)Proving $\lim_{x\to\infty}\frac1{x^s}\int_0^xt^{s-1}f(\sin t)dt=\frac1{2\pi s}\int_0^{2\pi}f(\sin t)dt$, for $s>0$ and $f$ defined on $[-1,1]$. Then, I realized that it was a specific case of the problem: What is $\lim_{t\rightarrow \infty }\int_{a}^{b}f(x,\sin(tx))dx$？ (Solved by @Gerd). I wonder if it can be generalized furthur.
I conjecture:
Let $f\in C(D\times S)$ where D is the integral domain ($D\subseteq\mathbb{R}^n$) and $S$ is the 'periodic' domain of parameters of the n-dimensional loop surface (close or repeat itself) that parametrized by $\phi(x_1,..,x_n)$, $\mu(S)$ is the volume of $S$; $c_1,..,c_n$ are bound variables:
$$\lim_{(c_1,..,c_n)\to(\infty,..,\infty)}\int_D f(x_1,..,x_n,\phi(c_1x_1,..,c_nx_n))\,dx_1..dx_n=$$ $$\frac{1}{\mu(S)}\int_{D\times S}f(x_1,..,x_n,\phi(y_1,..,y_n))\,dx_1..dx_ndy_1..dy_n$$
My attempt to explain why it is true:
Since $\phi$ is 'periodic' and $c_i\to\infty$ ($i=1,..,n$), ignore degeneracies, the probabilty that $\phi(c_1y_1,..,c_ny_n)$ land on each point on the surface is equally distributed, then $f$ tends to its expected value. As $(c_1,..,c_n)\to(\infty,..,\infty)$: $$\forall^{\infty} (x_1,..,x_n)\in D:f(x_1,..,x_n,\phi(c_1x_1,..,c_nx_n))\to$$ $$\frac{1}{\mu(S)}\int_S f(x_1,..,x_n,\phi(y_1,..,y_n))dy_1..dy_n$$ Gerd's solution is great but I feel the conjecture can be proved with my argument above but I can't make it rigorous though. I tried some examples on Desmos and it seems to support the conjecture (albeit the terrible convergence rate) Also, if this kind of problem was investigated somewhere long ago, please let me know the source.
Any help is appreciated.
