Question: Let \(f\) and \(g\) be the functions given by \(f(x)=1+3 \sin \left(\frac{\pi x}{2}\right)\) and \(g(x)=2^{-x} .\) Let \(R\) be the shaded region in the first quadrant enclosed by the graphs of \(f\) and \(g\) as shown in the figure below:

1. Find the area of region \(\mathrm{R}\).
2. Find the volume of the solid generated when \(\mathrm{R}\) is revolved about the \(x\) -axis.
3. Find the volume of the solid generated when the region \(\mathrm{R}\) is revolved about the line \(x=4\).
4. The region \(\mathrm{R}\) is the base of a solid. For this solid, the cross sections perpendicular to the \(x\) -axis are equilateral triangles with one side extending from \(y=f(x)\) to \(y=g(x)\). Find the volume of this solid.

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