Cylindrical Coordinates — Question 4

Question 4 – Solution

Strategy Substitute $x=r\cos\theta$ and $y=r\sin\theta$.

Equation $$\boxed{r(\cos\theta+\sin\theta)+z=4}.$$ If $\cos\theta+\sin\theta\ne 0$, then $$\boxed{r=\frac{4-z}{\cos\theta+\sin\theta}}.$$

Exceptional angles $\cos\theta+\sin\theta=0$ when $\theta=3\pi/4$ or $7\pi/4$ modulo $2\pi$. At those directions, the original cylindrical equation reduces to $z=4$; dividing by the zero angular factor would incorrectly discard or invent points.