Cylindrical Coordinates — Question 6

Question 6

Find a one-parameter vector representation for the curve where the cylinder $x^2+y^2=4$ meets the plane $z=x+y$. State an interval that traces the curve exactly once.

Question 6 – Solution

Strategy The cylinder fixes $r=2$; use $\theta=t$ and substitute into the plane.

Coordinates $$x=2\cos t,\qquad y=2\sin t,\qquad z=x+y=2\cos t+2\sin t.$$ Therefore $$\boxed{\vec r(t)=\left\langle 2\cos t,2\sin t,2\cos t+2\sin t\right\rangle}, \qquad 0\le t<2\pi.$$

Verification The first two components satisfy $x^2+y^2=4$, and the third is their sum. The half-open interval prevents duplicating the starting point.