Suppose I have a $2n-1$ dimensional region in $\mathbb{R}^{2n}$ and a function which are both particularly nice when written in polar coordinates $(r_1,\theta_1,r_2,\theta_2,\cdots,r_n,\theta_n)$: the region is defined by $\sum a_i\theta_i=c$ and the function by $\prod r_i^{b_i}$. I'm struggling to figure out what volume form I need to put on this region to correctly compute the integral.
For the $n=1$ case, things are sort of too easy to give me any idea of what's going on- the problem's an integral along a line. The next case, $n=2$, involves an integral over some 3-dimensional subspace of $\mathbb{R}^4$, which I find it extremely hard to visualize and see what the correct volume form is- I know I should have some factor of $r_1$ and maybe of $r_2$ to account for some of the rotation I can make in the region- this leads me to make the guess that the general answer should be something like $r_1r_2\cdots r_{n-1} dr_1\cdots dr_n d\theta_1\cdots d\theta_{n-1}$, but I'm kind of lost right now and would appreciate some course correction if you could give it.
Your guess is right. Represent your equation for the region (possibly after renumbering the coordinates) as $$ \theta_n=t_0+\sum_{j=1}^{n-1}a_j\theta_j. $$ Consider the coordinates $(r_1,\theta_1,\dotsc,r_{n-1},\theta_{n-1},r_n,t)$ on $\mathbb{R}^{2n}$, where $$ t=\theta_n-\sum_{j=1}^{n-1}a_j\theta_j. $$
The volume form on $\mathbb{R}^{2n}$ is $$ dvol=r_1\dots r_{n-1}r_n dr_1\dotsm dr_{n-1}dr_n d\theta_1\dotsm d\theta_{n-1}d\theta_n\\=\bigl(r_1\dots r_{n-1} dr_1\dotsm dr_{n-1}dr_n d\theta_1\dotsm d\theta_{n-1}\bigr)\bigl(r_ndt\bigr). $$ Now the second factor, $r_ndt$, is the arc length along the circle orthogonal to your region $\{t=t_0\}$, and hence the first factor is teh volume form on the region itself.