How to write the limits of triple integral $\iiint f(x,y,z) dz dy dx$ over an annulus which lies between the circle of radii $r$ and $R$, $r<R$? I am confused. I don't want to change into polar coordinates.
EDIT: I made a mistake in asking my question. It should be the spherical shell region lying between two spheres of radius $r$ and $R$. Sorry!
In polar coordinates:
$$\int\limits_{r=r_i}^R \int\limits_{\theta = 0}^{2 \pi} \int\limits_{z=z_i}^{z_f} f(r \cos \theta, r \sin \theta,z)\ r\ dr\ d \theta\ dz$$
or in rectilinear coordinates...
$$\int\limits_{x=-R}^R\ dx \int\limits_{y = - \sqrt{R^2 - x^2}}^{+ \sqrt{R^2 - x^2}}\ dy \int\limits_{z=z_i}^{z_f}\ dz\ f(x,y,z) - \int\limits_{x=-r}^r\ dx\ \int\limits_{y = - \sqrt{r^2 - x^2}}^{+ \sqrt{r^2 - x^2}}\ dy\ \int\limits_{z=z_i}^{z_f}\ dz\ f(x,y,z)$$
Revised question:
$$\int\limits_{x=-R}^R\ dx \int\limits_{y = -\sqrt{R^2 - x^2 - z^2}}^{+\sqrt{R^2 - x^2 - z^2}}\ dy \int\limits_{z=-\sqrt{R^2 - x^2 - y^2}}^{+\sqrt{R^2 - x^2 - y^2}}\ dz\ f(x,y,z) - \int\limits_{x=-r}^r\ dx \int\limits_{y = -\sqrt{r^2 - x^2 - z^2}}^{+\sqrt{r^2 - x^2 - z^2}}\ dy \int\limits_{z=-\sqrt{r^2 - x^2 - y^2}}^{+\sqrt{r^2 -x^2 - y^2}}\ dz\ f(x,y,z)$$