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Describe the set in cylindrical coordinates:

A = {(x,y,z) ∈ R3 : y^2 + z^2 ≤ 4, |x|≤1}

My solution: We use the cylindrical coordinates r,θ,z.

x,y,z expressed in cylindrical coordinates in this case: x=x, y =r sin(θ), z=r cos(θ). But in this case θ angle is measured clockwise from the positive z-axis.

Then the set in cylindrical coordinates would be described as:

-2 ≤ r ≤ 2,

0 ≤ θ ≤ 2π,

z=r cos(θ).

This feels weird since I'm changing the meaning of θ from the "standard interpretation" (where θ is measured clockwise from the positive x-axis). Or I'm I supposed to use the cylindrical coordinates r,θ,x? I feel lost.

How should I solve this?

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1 Answer 1

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That's almost correct. Remember that $r$ is a distance; therefore, it cannot be smaller that $0$. And you forgot to bound the values of $x$. So, it should be:$$\left\{\begin{array}{l}-1\leqslant x\leqslant1\\0\leqslant r\leqslant2\\0\leqslant\theta\leqslant2\pi.\end{array}\right.$$

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  • $\begingroup$ So we use the cylindrical coordinates (r,θ,x) and x=x, y =r sin(θ), z=r cos(θ), right? But should I specify how θ is measured? In my textbook, θ is measured counter clockwise from the positive x-axis (in the xy-plane), but here it is measured counter clockwise from the positive z-axis (in the zy-plane). @José $\endgroup$
    – user1
    Commented Jan 21, 2022 at 14:14
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    $\begingroup$ It makes no difference. And actually I was thinking that you had chosen $y=r\cos\theta$ and $z=r\sin\theta$. $\endgroup$ Commented Jan 21, 2022 at 14:47
  • $\begingroup$ Yes I made a mistake, It should be 𝑦 = 𝑟 cos 𝜃 and 𝑧 = 𝑟 sin 𝜃. Thanks for helping me! :)@José $\endgroup$
    – user1
    Commented Jan 21, 2022 at 15:23

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