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?

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?

Question
Alternate coordinate systems
asked 2021-01-06
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?

Answers (1)

2021-01-07
Step 1
The equation relating to rectangular (x, y, z) and cylindrical \(\displaystyle{\left({r},\theta,{z}\right)}\) coordinates are, \(\displaystyle{x}={r} \cos{\theta},{y}= \sin{\theta},{z}={z}\)
\(\displaystyle{r}^{2}={x}^{2}+{y}^{2}\)
Here, \(\displaystyle \tan{\theta}=\frac{y}{{x}}\)
Hence , the solution is \(\displaystyle \tan{\theta}=\frac{y}{{x}}.\)
The equation relating spherical coordinates to Cartesian and cylindrical coordinates is, \(\displaystyle{r}=\delta \sin{\theta},{x}={r} \cos{\theta}=\delta \sin{\cancel{\circ}} \cos{\theta}\)
\(\displaystyle{z}=\delta \cos{\cancel{\circ}},{y}={r} \sin{\theta}=\delta \sin{\cancel{\circ}} \sin{\theta}\)
Step 2
Here, \(\displaystyle\delta=\sqrt{{{x}^{2}+{y}^{2}+{z}^{2}}}\)
From equation 1, we know \(\displaystyle{x}^{2}+{y}^{2}={r}^{2}.\)
Therefore,
\(\displaystyle\delta=\sqrt{{{r}^{2}+{z}^{2}}}\)
Hence, the solution is \(\displaystyle\delta=\sqrt{{{r}^{2}+{x}^{2}}}.\)
The diagram regarding the cylindrical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) is given below:
image
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