Answer:
The equation for z for the parametric representation is [tex]z = 7 \sin (v)[/tex] and the interval for u is [tex]0\le u\le 1[/tex].
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 [tex]y^2+z^2 = 49[/tex], that lies between the places x = 0 and x = 1.
[tex]x=u\\ y= 7 \cos(v)\\z=? \\ 0\le v\le 2\pi \\ ?\le u\le ?[/tex]
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
[tex]0\le u\le 1[/tex]
Equation for z.
Replacing the given equation for the parameterization [tex]y = 7 \cos(v)[/tex] on the given equation for the cylinder give us
[tex](7 \cos(v))^2 +z^2 = 49 \\ 49 \cos^2 (v)+z^2 = 49[/tex]
Solving for z, by moving [tex]49 \cos^2 (v)[/tex] to the other side
[tex]z^2 = 49-49 \cos^2 (v)[/tex]
Factoring
[tex]z^2 = 49(1- \cos^2 (v))[/tex]
So then we can apply Pythagorean Theorem:
[tex]\sin^2(v)+\cos^2(v) =1[/tex]
And solving for sine from the theorem.
[tex]\sin^2(v) = 1-\cos^2(v)[/tex]
Thus replacing on the exercise we get
[tex]z^2 = 49\sin^2 (v)[/tex]
So we can take the square root of both sides and we get
[tex]z = 7 \sin (v)[/tex]
