Cylindrical Coordinates — Question 2

Question 2

Rewrite the surface $x^2+y^2=4x$ in cylindrical coordinates. Identify the surface geometrically, including its axis and radius.

Question 2 – Solution

Strategy Replace $x^2+y^2$ by $r^2$ and $x$ by $r\cos\theta$.

Cylindrical equation $$r^2=4r\cos\theta.$$ The equation $r=4\cos\theta$ describes the non-axis points; $r=0$ is already included, so $$\boxed{r=4\cos\theta}.$$

Geometry In rectangular form, $$(x-2)^2+y^2=4.$$ Thus the surface is a vertical circular cylinder of radius $2$, whose axis is the line $\boxed{x=2,\ y=0}$ parallel to the $z$-axis.