Let:
\begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align}
Consider the expression:
\begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align}
We next have to change the variables from $(r,s,t)$ to $(\theta, a, p)$
The Jacobian of the coordinate transformation (after computing) is:
$J= \dfrac{\partial r}{\partial \theta}=\dfrac{a\ p\ \sin\theta}{r}$
Thus our new function becomes $J\ f(r) =\dfrac{a\ p\ \sin\theta}{r} f(r)$
Question:
One of my friends said that the limits of the integration would be as follows:
\begin{align} \int_{q'}^q \int_{b'}^b g(a)\ \int_{0}^{\pi} \dfrac{a\ p\ \sin\theta}{r} f(r) \ d \theta\ da\ dp\ \end{align}
Is he correct?