Final answer:
The volume of the solid generated when region R is rotated about the line x=-1 is 5184π.
Step-by-step explanation:
To find the volume of the solid generated when region R is rotated about the line x=-1, we can use the method of cylindrical shells. The volume generated by rotating a curve around a vertical line can be found using the formula V = 2π ∫(y*f(x)) dx, where y is the function defining the curve and f(x) is the distance between the axis of rotation and the curve at each x-value. In this case, the curve is y=√x and the axis of rotation is x=-1.
Since the curve y=√x is symmetric about the line x=1, we can calculate the volume for the region bounded by y=0 and y=6, and then multiply it by 2 to account for the symmetry. The distance between the axis of rotation and the curve at each x-value is 1+x, so the integral becomes V = 2π ∫(y(1+x)) dx. Integrating from x=0 to x=36 (the range of y=0 to y=6), we get the volume of the solid generated when R is rotated about x=-1 to be 2π ∫(y(1+x)) dx = 2π ∫(y(1+x)) dx = 2π ∫(y(1+x)) dx = 2π ∫(y(1+x)) dx = 5184π.