Question: (Tubular shape). Let \(C\) be the cycloid given by the vector-valued function

in the three-dimensional space \(\mathbf{R}^{3}\). Consider the unit disk \(D\) in the xy-plane in \(\mathbf{R}^{3}\), i.e.

Let this disk move in the space in the following way: the center moves along the cycloid \(C\) for \(0 \leq t \leq 2 \pi\), and at each moment the disk is perpendicular to the unit tangent vector of the cycloid. The trajectory of this moving disk is a solid tubular region \(T\) around $C .$ Let \(S\) denote the boundary surface of this solid region except the initial and final disks, i.e. \(S\) is like a twisted tube

1. Give parametric equations for \(S\) and for \(T\).
2. Use (a) to write down a double integral for the surface area of \(S\). But, do not compute this area.
3. Use (a) to write down a triple integral for the volume of \(T\). But, do not compute this volume.
4. Use (a) to evaluate the surface integral

where \(\mathbf{F}(x, y, z)=<0,0, x>\) and the surface \(S\) is oriented with outward normal vector.

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