$(9$ points$)$ Let $R$ be the region bounded by the line $x=2,$ the line $y=4$ and the curve $xy=4.$ Let $V$ be the solid obtained by revolving the region $R$ around the $y-$axis.

$($a$)$ Sketch the region $R.$

$($b$)$ Use the washer method to write the integral that gives the volume of $V$ and evaluate the integral.

$($c$)$ Use the shell method to write the integral that gives the volume of $V$ evaluate the integral.

$($d$)$ An artist would like to paint the region $R$ using different shades of the color green. Suppose that the density is given by $(s),$ where $s$ represents the distance from the line $x=2.$ Write a Riemann sum approximating the total amount of paint needed. Your final answer should be written in terms of $x,x_k,$ please make sure to explain exactly what your notations mean.