I have this doubt about spherical and cylindrical coordinates transformations. When we have to integrate some function over some region, when we substitute with a linear transformation, we are changing the shape of the region into one that is more convenient. The same goes for spherical and cylindrical coordinate transformations. If we are integrating over a cube for example, when we change our coordinate system into spherical coordinates the cube gets transformed into a spherical cube like this one:
And something similar happens with cylindrical coordinates. But what actually happens when we are integrating over a sphere and apply a substitution to go from Cartesian to spherical? Does the sphere shape gets preserved? If so, how do you demonstrate that the form of the region hasn't changed? It gets expanded or rotated?
Consider this region for example $x²+y²+z² \le 9$ and $z \ge 2$ which looks like the cover of a jar. How does it get transformed when we make the substitution into spherical coordinates?
When you change the coordinate system then your original region of integration shape gets changed to another. For example in the case of a sphere. If you change from cartesian to spherical coordinates the sphere is changed into a $3D$ rectangle in spherical coordinates domain. That's the whole point of employing the coordinate transformation: The shape becomes simpler in your other domain. Why does it become a rectangle? Because to calculate the volume of the sphere you integrate $\theta$ from $0$ to $\pi$, $\phi$ goes from $0$ to $2 \pi$, and $r$ goes from $0$ to $R$ where $R$ is the radius of the sphere. Of course you have an extra factor in the integrand coming from the determinant of the Jacobian of transformation (which accounts for local change of volume due to the transformation) but the domain of integration is much simpler than the complicated geometry of a sphere in cartesian coordinates.