determination of a volume of a solid being lifted

Question

i passed by a question now and wanted to check something, i want to calculate the volume of a given solid such that the solid is representedd by K= ${ (x,y,z)\in R^3 / x^2 +y^2 \le \frac{1}{z^2}}$ and ${1\lt z \lt 3 }$ my question is using cylindrical coordinates $r$ varies between {0 and $\frac{1}{z}$} or {1 and $\frac{1}{z}$}?

Answer 1

Setting $r^2 = x^2 + y^2$ you get the two following constrains : $$ r^2 \leq \frac{1}{z^2} \overset{(*)}{\Leftrightarrow} r \leq \frac{1}{z}\\ 1 < z < 3 $$ $(*)$ Since $r \geq 0$ with cylindrical coordinates.

Therefore $r$ ranges in $r \in [0,1/z]$ for a given fixed $z$ because we have an upper bound on $r$ but no (explict) lower bound.