 # How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates? nicekikah 2021-08-01 Answered
How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

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Step 1
Give the notes about how the triple integrals defined in cylindrical and spherical coordinates.
Step 2
The cylindrical coordinates denotes a point P in space by ordered triples $$\displaystyle{\left({r},\theta,{z}\right)}$$ in that r and $$\displaystyle\theta$$ are polar coordinates for the vertical projection of P on the xy-plane with $$\displaystyle{r}\geq\theta$$ and z is the rectangular vertical coordinate.
The equations related to the rectangular coordinates $$\displaystyle{\left({x},{y},{z}\right)}$$ and cylindrical coordinates $$\displaystyle{\left({r},\theta,{z}\right)}$$ are,
$$\displaystyle{x}={r}{\cos{\theta}},{y}={r}{\sin{\theta}},{z}={z},{r}^{{{2}}}={x}^{{{2}}}+{y}^{{{2}}}$$ and $$\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}$$
Step 3
The spherical coordinates represent a point P in space 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.
Step 4
The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,
$$\displaystyle{r}=\rho{\sin{\phi}}$$
$$\displaystyle{x}={r}{\cos{\theta}}=\rho{\sin{\phi}}{\cos{\theta}}$$
$$\displaystyle{z}=\rho{\cos{\phi}}$$
$$\displaystyle{r}{\sin{\theta}}=\rho{\sin{\phi}}{\sin{\theta}}$$
$$\displaystyle\rho=\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}+{z}^{{{2}}}}}$$
$$\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}$$
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.