I am looking for literature on the type of multidimensional integral below
$$F(a_1,...,a_D)=\int_{Q} f(q_1,...,q_D) \prod_{d=1}^D[(a_d)^{q_d}] d^{D-1}q $$
The integral domain Q is a D-1 dimensional hyperplane within a D-Euclidean space for which $$ q_1+...+q_D=1 $$
I'm sorry in advance if this was asked before (I searched and couldn't find posts on this topic), or if my question is too general. This popped-up while working on field theory and I don't even know what keywords to search in Google... Any information will help, like integrability conditions, known identities, etc.
For clarity, in two dimensions this will be a line integral
$$F(a,b)=\sqrt 2 \int_{-\infty}^{\infty} f(q) a^q b^{1-q} dq $$