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# 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? # 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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Alternate coordinate systems asked 2020-12-28
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?

## Answers (1) 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.

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