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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hesgidiauE
Answered 2021-08-02 Author has 16664 answers
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.
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