# 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?

Question
Alternate coordinate systems
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?

2020-12-29

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 $$\displaystyle{\left({r},\theta,{z}\right)}\in{t}\hat{{r}}{\quad\text{and}\quad} \theta$$ 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 (x, y, z) 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 $$\displaystyle{\left({p},\phi,\theta\right)}$$ in which, p is the distance from P to the origin $$(p \geq 0)$$ $$\displaystyle\phi$$ is 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 ctlindrical coordinates are, $$\displaystyle{r}={p}{\sin{\phi}}$$
$$\displaystyle{x}={r}{\cos{\theta}}={p}{\sin{\phi}}{\cos{\theta}}$$
$$\displaystyle{z}={p}{\cos{\phi}}$$
$$\displaystyle{y}={r}{\sin{\theta}}={p}{\sin{\phi}}{\sin{\theta}}$$
$$\displaystyle{p}=\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}+{z}^{{{2}}}}}$$
$$\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}$$ 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 constant coordinate value.

### Relevant Questions

How are triple integrals defined in cylindrical and spherical coordinates?
Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?
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.
To compare and contrast: the rectangular, cylindrical and spherical coordinates systems.
Consider the solid that is bounded below by the cone $$z = \sqrt{3x^{2}+3y^{2}}$$
and above by the sphere $$x^{2} +y^{2} + z^{2} = 16.$$.Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.
What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?
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a. rectangular
b. polar
c. cylindrical
d. spherical
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