Find a parametric representation for the part of the cylinder y2 + z2 = 4 that lies between the planes x = 0 and x = 5.
x = u
y = 2cos(v)
z =
0 ≤ v ≤ 2π
To find a parametric representation for the part of the cylinder y^2 + z^2 = 4 that lies between the planes x = 0 and x = 5, we can use cylindrical coordinates, where x = u, y = rcos(v), and z = rsin(v).
Notice that in cylindrical coordinates, the equation y^2 + z^2 = 4 represents a circular cylinder with radius 2, centered at the origin.
We are given that 0 ≤ v ≤ 2π, which means v ranges from 0 to 2π, completing a full revolution around the cylinder.
Now, we need to determine the range of u. Since the cylinder lies between the planes x = 0 and x = 5, we have 0 ≤ u ≤ 5.
To summarize, the parametric representation for the part of the cylinder between the planes x = 0 and x = 5 is:
x = u
y = 2 cos(v)
z = 2 sin(v)
where 0 ≤ u ≤ 5 and 0 ≤ v ≤ 2π.