Let $\xi$ random vector and $\zeta^{x}:=x^{T}\xi$ random variable. Is it correct to say that $\int x^{T}y f_{\xi}(y)dy=\int tf_{\zeta^{x}}(t)dt$? Let $\xi\in\mathbb{R}^{m}$ be a random vector with  joint density function $f_{\xi}:\mathbb{R}^{m}\rightarrow \mathbb{R}$. Let $x\in \mathbb{R}^{x}$ be a nonzero vector. We consider the random variable $\zeta^{x}:=\langle x,\xi\rangle=x^{T}\xi$, we called $f_{\zeta^{x}}:\mathbb{R}\rightarrow\mathbb{R}$   the density function of $\zeta^{x}$.

The question: Is it correct to say that $$\int_{\mathbb{R}}\cdots\int_{\mathbb{R}} \langle x,y \rangle f_{\xi}(y)dy_{1}\cdots dy_{m}=\int_{\mathbb{R}}tf_{\zeta^{x}}(t)dt  .?$$
Yes, you can write $E[g(\xi)]$ as either $\int_{\mathbb{R}^m} g(y) f_\xi(y) \, dy$ or $\int_{\mathbb{R}} t f_{g(\xi)}(t) \, dt$. The first is the "Law of the unconscious statistician"
while the second is one definition of the expectation of a random variable.
Yes, you can write $E[g(\xi)]$ as either $\int_{\mathbb{R}^m} g(y) f_\xi(y) \, dy$ or $\int_{\mathbb{R}} t f_{g(\xi)}(t) \, dt$. The first is the "Law of the unconscious statistician"
while the second is one definition of the expectation of a random variable.