Cylindrical Coordinates — Question 5

Question 5 – Solution

Strategy Complete the square in $z$ after replacing $x^2+y^2$ by $r^2$.

Equation $$r^2+z^2=6z \quad\Longleftrightarrow\quad \boxed{r^2+(z-3)^2=9}.$$

Geometry This is a sphere of radius $3$ centered at $(0,0,3)$. It is independent of $\theta$, reflecting rotational symmetry about the $z$-axis.

Ranges Since $r^2\le 9$, $\boxed{0\le r\le 3}$. The vertical extent is $\boxed{0\le z\le 6}$. At a fixed height $z$, the horizontal trace has radius $\sqrt{9-(z-3)^2}$.