Question

Give a parametric description for a cylinder with radius \(a\) and height \(h,\) including the intervals for the parameters.

Short answer

Question: Provide the parametric description of a cylinder with radius \(a\) and height \(h\), including the intervals for the parameters. Answer: The parametric description of the cylinder is \((x,y,z) = (a\cos(u), a\sin(u), v)\), with \(u \in [0, 2\pi]\) and \(v \in [0,h]\).

Step-by-step solution

Express the points on the surface of the cylinder in terms of parameters \(u\) and \(v.\)

To express the points on the surface of the cylinder, we need to use the cosine and sine functions, which will give us the varying x and y coordinates as we go around the circle centered at the origin. The z-coordinate, or the height of the cylinder, can be represented by parameter \(v.\) Considering that the cylinder is aligned with the z-axis: - The x-coordinate can be expressed as \(a \cos(u)\). - The y-coordinate can be expressed as \(a \sin(u)\). - The z-coordinate can be expressed as \(v\). Therefore, the parametric description of a point on the cylinder surface is: \((x,y,z) = (a\cos(u), a\sin(u), v)\)

Determine the intervals for parameters \(u\) and \(v.\)

Parameter \(u\) represents the angle around the cylinder's axis, so it will vary from \(0\) to \(2\pi\), because to cover the whole circumference, a complete circle is needed. \(u \in [0, 2\pi]\) Parameter \(v\) represents the height of the cylinder. Since it has a height of \(h\), \(v\) will vary from \(0\) (bottom) to \(h\) (top). \(v \in [0,h]\)

Write the final parametric description with intervals for the parameters.

The parametric description of the cylinder with radius \(a\) and height \(h\) is: \((x,y,z) = (a\cos(u), a\sin(u), v)\) with \(u \in [0, 2\pi]\) and \(v \in [0,h]\).