I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\mapsto(x,y,z);J_g=r^2\sin\theta.\implies V=\int_{0}^{2\pi}d\varphi \int_{0}^{\pi}\sin\theta d\theta \int_{0}^{r}r^2dr$$ Can I do something similar with a torus? Like this?Since the large Radius is fixed?
$$x=(R+r\cos\theta)\cos \varphi , y=(R+r\cos\theta)\sin \varphi, z=r\sin \theta \\ g(r,\varphi,\theta)\mapsto(x,y,z);J_g=R+r\cos\theta\implies V=\int_{0}^{2\pi}d\varphi \int_{0}^{2\pi} d\theta \int_{0}^{r}(R+r\cos\theta) dr\ \ ??$$
The integral has serious problems (the bounds of integration don't define the torus in spherical coordinates), but the volume is well-known to be $(2\pi R)(\pi r^{2})$ by Pappus' theorem. If you want to see this with integration, cylindrical coordinates (or the shell method from one-variable calculus) are easier than spherical.
(As written, incidentally, the volume integral for the sphere isn't quite OK, either: You're using $r$ to denote both the radius function and the fixed radius of the sphere whose volume you're calculating.)