In this case, we need a Jacobi determinant $$\begin{equation*} \prod_{i=1}^d (b_i-a_i), \end{equation*}$$ and to convert the function $g(x_1,\dots,x_d)$ to $$\begin{equation*} g(x_1,\dots,x_d)\rightarrow g(a_1+(b_1-a_1)t_1,\dots,a_d+(b_d-a_d)t_d). \end{equation*}$$