Question

Find a parametric representation for the surface.

The part of the cylinder
y² + z² = 81
that lies between the planes x = 0 and x = 5. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.)

Answer 1

The parametric equations x = u, y = 9cos(v), z = 9sin(v) with u ranging from 0 to 5 and v from 0 to 2π represent the surface of the cylinder y² + z² = 81 between the planes x = 0 and x = 5.

Step-by-step explanation:

To find a parametric representation for the surface of the part of the cylinder y² + z² = 81 that lies between the planes x = 0 and x = 5, we can use the following parametric equations:

• x = u
• y = 9cos(v)
• z = 9sin(v)

Here, the parameter u will vary between 0 and 5 since the cylinder lies between these two planes along the x-axis. The parameter v will vary from 0 to 2π (or 0 to 360 degrees) to describe a full circle in the yz-plane. This effectively covers the entire surface of the given part of the cylinder.