Final answer:
To find the volume of the solid generated by revolving the region, we use the shell method and integrate the formula 2πrh from the lower limit to the upper limit. The final answer is 64π/3 cubic units.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by y = √x, the x-axis, and the line x = 4 about the y-axis using the shell method, we can divide the region into infinitesimally thin cylinders. The volume of each cylinder is given by the formula V = 2πrh, where r is the radius and h is the height. In this case, the radius is given by x = 4 - y² and the height is given by the difference between the x-values of the curve and the line x = 4. To find the total volume, we integrate the formula from the lower limit to the upper limit.
∫(from 0 to 4) 2π (4-y²) (4-0) dy
simplifying the equation gives:
∫(from 0 to 4) 8π (4-y²) dy
which equals 8π * (4y - (y³/3))
evaluating the integral gives the final answer:
64π/3 cubic units.