Given curves are y=18x, y=18, and x=0. We have to find the volume of the solid generated when R is revolved about the y-axis using the shell method.Let's sketch the curves first:We can see that R is the region bounded by the curves y=18x, y=18 and x=0.
We need to rotate this region around the y-axis using the shell method to find the volume of the solid generated.Let's use the shell method here:Shell method:$$\large V=2\pi \int_{a}^{b}x[f(x)-g(x)]dx$$
Here, we need to rotate R around the y-axis, therefore, a = 0 and b = 18.
From the graph, we can see that the functions $f(x)$ and $g(x)$ are as follows:$$f(x) = 18 \\ g(x) = 18x$$
Using the shell method, the volume of the solid generated is:$$\begin{aligned}V &= 2\pi \int_{0}^{18}x[18-18x]dx \\&= Therefore, the volume of the solid generated when R is revolved about the y-axis is 17496π.