Question # Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.

Alternate coordinate systems
ANSWERED Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates. 2021-02-06
Cylindrical coordinates prefer represent a point P in space by ordered triples $$\displaystyle{\left({r},\theta,\ {z}\right)}$$ in which r and $$\displaystyle\theta$$ are polar coordinates for the vertical projection of P on the $$\displaystyle{x}{y}={p}{l}{a}\ne,$$ with $$\displaystyle{r}\ \geq\ {0}$$, and x is the rectangular vertical coordinate.
The equations relating rectangular $$\displaystyle{\left({x},\ {y},\ {z}\right)}$$ and cylindrical $$\displaystyle{\left({r},\theta,\ {z}\right)}$$ coordinates are,
$$\displaystyle{x}={r}{\cos{\theta}}$$
$$\displaystyle{y}={r}{\sin{\theta}}$$
$$\displaystyle{z}={z}$$
$$\displaystyle{r}^{{{2}}}={x}^{{{2}}}\ +\ {y}^{{{2}}}$$
$$\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}$$
The spherical coordinates represent a point P in by ordered triples $$\displaystyle{\left(\rho,\phi,\theta\right)}$$ in which,
$$\displaystyle\rho$$ is the distance from P to the origin $$\displaystyle{\left(\rho\ \geq\ {0}\right)}$$
. $$\displaystyle\phi$$ is the angle $$\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{O}{P}\right\rbrace}$$ makes with the positive z - axis $$\displaystyle{\left({0}\ \leq\ \phi\ \leq\ \pi\right)}$$
. $$\displaystyle\theta$$ is the angle from cylindrical coordinates.
The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,
$$\displaystyle{r}=\rho{\sin{\phi}}$$
$$\displaystyle{x}={r}{\cos{\theta}}=\rho{\sin{\phi}}{\cos{\theta}}$$
$$\displaystyle{z}=\rho{\cos{\phi}}$$
$$\displaystyle{y}={r}{\sin{\theta}}=\rho{\sin{\phi}}{\sin{\theta}}$$
$$\displaystyle\rho=\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}\ +\ {z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}\ +\ {z}^{{{2}}}}}$$
$$\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}$$
Cylindrical coordinates are good for describing cylinders whose axes run along the z - axis and planes that either contain the z - axis or lie perpendicular to the z -axis.
Surfaces like these have equations of constant coordinate value.