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?

ossidianaZ 2021-01-06 Answered
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?
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Expert Answer

Bertha Stark
Answered 2021-01-07 Author has 96 answers

Step 1
The equation relating to rectangular (x, y, z) and cylindrical (r,θ,z) coordinates are, x=rcosθ,y=sinθ,z=z
r2=x2+y2
Here, tanθ=yx
Hence , the solution is tanθ=yx.
The equation relating spherical coordinates to Cartesian and cylindrical coordinates is, Undefined control sequence \cancel
Undefined control sequence \cancel
Step 2
Here, δ=x2+y2+z2
From equation 1, we know x2+y2=r2.
Therefore,
δ=r2+z2
Hence, the solution is δ=r2+x2.
The diagram regarding the cylindrical coordinates (r,θ,z) is given below:
image

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