Suppose we have the general uniform distribution $$\begin{equation*} p(y)dy=\left\{\begin{array}{cc} \frac{dy}{b-a} & a \le y \le b\\ 0 & else\end{array}\right. \end{equation*}$$ If we wish to relate this distribution to the one in the interval $x \in [0,1]$ we have $$\begin{equation*} p(y)dy=\frac{dy}{b-a}=dx, \end{equation*}$$ and integrating we obtain the cumulative function $$\begin{equation*} x(y)=\int_a^y \frac{dy'}{b-a}, \end{equation*}$$ yielding $$\begin{equation*} y=a+(b-a)x, \end{equation*}$$ a well-known result!