Answer:
The equation for z for the parametric representation is
and the interval for u is
.
Step-by-step explanation:
You have the full question but due lack of spacing it looks incomplete, thus the full question with spacing is:
Find a parametric representation for the part of the cylinder
, that lies between the places x = 0 and x = 1.
![x=u\\ y= 7 \cos(v)\\z=? \\ 0\le v\le 2\pi \\ ?\le u\le ?](https://tex.z-dn.net/?f=x%3Du%5C%5C%20y%3D%207%20%5Ccos%28v%29%5C%5Cz%3D%3F%20%5C%5C%200%5Cle%20v%5Cle%202%5Cpi%20%5C%5C%20%3F%5Cle%20u%5Cle%20%3F)
Thus the goal of the exercise is to complete the parameterization and find the equation for z and complete the interval for u
Interval for u
Since x goes from 0 to 1, and if x = u, we can write the interval as
![0\le u\le 1](https://tex.z-dn.net/?f=0%5Cle%20u%5Cle%201)
Equation for z.
Replacing the given equation for the parameterization
on the given equation for the cylinder give us
![(7 \cos(v))^2 +z^2 = 49 \\ 49 \cos^2 (v)+z^2 = 49](https://tex.z-dn.net/?f=%287%20%5Ccos%28v%29%29%5E2%20%2Bz%5E2%20%3D%2049%20%5C%5C%2049%20%5Ccos%5E2%20%28v%29%2Bz%5E2%20%3D%2049)
Solving for z, by moving
to the other side
![z^2 = 49-49 \cos^2 (v)](https://tex.z-dn.net/?f=z%5E2%20%3D%2049-49%20%5Ccos%5E2%20%28v%29)
Factoring
![z^2 = 49(1- \cos^2 (v))](https://tex.z-dn.net/?f=z%5E2%20%3D%2049%281-%20%5Ccos%5E2%20%28v%29%29)
So then we can apply Pythagorean Theorem:
![\sin^2(v)+\cos^2(v) =1](https://tex.z-dn.net/?f=%5Csin%5E2%28v%29%2B%5Ccos%5E2%28v%29%20%3D1)
And solving for sine from the theorem.
![\sin^2(v) = 1-\cos^2(v)](https://tex.z-dn.net/?f=%5Csin%5E2%28v%29%20%3D%201-%5Ccos%5E2%28v%29)
Thus replacing on the exercise we get
![z^2 = 49\sin^2 (v)](https://tex.z-dn.net/?f=z%5E2%20%3D%2049%5Csin%5E2%20%28v%29)
So we can take the square root of both sides and we get
![z = 7 \sin (v)](https://tex.z-dn.net/?f=z%20%3D%207%20%5Csin%20%28v%29)