Find parametric equations for the cylinder $y ^ { 2 } + z ^ { 2 } = 16$
A) $\vec { r } ( s , \theta ) = 4 \cos \overrightarrow { \theta i } + 4 \sin \theta \vec { j }$
B) $\vec { r } ( s , \theta ) = 4 \cos \theta \vec { i } + 4 \sin \theta \vec { j } + s \vec { k }$
C) $\vec { r } ( s , \theta ) = \vec { s } + 16 \cos \theta \vec { j } + 16 \sin \theta \vec { k }$
D) $\vec { r } ( s , \theta ) = \operatorname { si } + 4 \cos \theta \vec { j } + 4 \sin \theta \vec { k }$
E) $\vec { r } ( s , \theta ) = \operatorname { si } + 4 \cos \theta \vec { j } + 4 \cos \theta \vec { k }$

Question

Quizplus · Accepted Answer

The given equation $y^2 + z^2 = 16$ describes a cylinder with radius 4 centered along the x-axis. The parametric equations for a circle in the yz-plane with radius 4 are $y = 4\cos\theta$ and $z = 4\sin\theta$. The x-coordinate is not constrained by the equation of the cylinder, so it can be any real number, represented by $s$. Thus, the correct parametric equations are $x = s$, $y = 4\cos\theta$, and $z = 4\sin\theta$, which corresponds to choice D: $\vec{r}(s, \theta) = s\vec{i} + 4\cos\theta\vec{j} + 4\sin\theta\vec{k}$.

Find Parametric Equations for the Cylinder y2+z2=16y ^ { 2 } + z ^ { 2 } = 16y2+z2=16

Find Parametric Equations for the Cylinder $y ^ { 2 } + z ^ { 2 } = 16$