# Describe set in cylindrical coordinates

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?

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:$$\left\{\begin{array}{l}-1\leqslant x\leqslant1\\0\leqslant r\leqslant2\\0\leqslant\theta\leqslant2\pi.\end{array}\right.$$
• It makes no difference. And actually I was thinking that you had chosen $y=r\cos\theta$ and $z=r\sin\theta$. Commented Jan 21, 2022 at 14:47