Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function $y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4$ about the $x$-axis.
A) $x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ { \frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$
B)$x = u , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ {\frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$
C)$x = u , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , 0 \leq u \leq 4 , v \geq 0$
D)$x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , u \geq 4,0 \leq v \leq 2 \pi$
E)$x = v , y = v ^ { \frac { 3 } { 7 } } \cos u , z = v ^ { \frac { 3 } { 7 } } \sin u , u \geq 4 , v \geq 0$

Question

Quizplus · Accepted Answer

When we revolve the graph of the function $y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4$ about the $x$-axis, we obtain a surface of revolution that looks like a vase or a wine glass. If we look closely, we can see that the radius of the vase at each height $y$ is given by $r = \left( y ^ { \frac { 7 } { 3 } } \right)^{\frac { 1 } { 2 } } = y ^ { \frac { 7 } { 6 } } $. Therefore, a good choice for parametric equations for this surface of revolution are:

$x = u , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ {\frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$

Here, we let $u$ represent the height of the vase, with $0 \leq u \leq 4$, and $v$ represent the angle of rotation around the $x$-axis, with $0 \leq v \leq 2 \pi$. Then, for each value of $u$, the point $(x,y,z)$ traces out a circle of radius $u ^ { \frac { 3 } { 7 } } $ around the $z$-axis. The angle $v$ determines the position of the circle in space. These parametric equations give us a smooth and well-defined surface of revolution that captures the shape of the vase. Therefore, the best choice is B.

Write a Set of Parametric Equations for the Surface of Revolution