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?
$\endgroup$1 Answer
$\begingroup$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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