Am I using the right limits on this triple integral?

[oliverkahn]

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?

 

[mathman]

If you look at it geometrically, r, a, p are lengths of three sides of a triangle, with r opposite $\theta$. The limits on r are for the two other sides subtracting or adding, which is equivalent to $\theta$ being 0 or $\pi$. Your friend is correct.