Final answer:
The volume of the solid obtained by rotating the region bounded by the curves 3x = y², x = 0, y = 5 about the y-axis can be found using the disk method and integration.
Step-by-step explanation:
The given question involves finding the volume of a solid of revolution generated by rotating the region bounded by the curve 3x = y², the line x = 0, and the line y = 5 about the y-axis. This can be accomplished by using the disk method, where the volume is calculated by integrating the area of circular disks along the axis of rotation.
The first step is to rewrite the equation in terms of y to find the radius of the disks. The region bounded by 3x = y² gives us the radius where x = √(y²/3). Now, we set up the integral from y=0 to y=5 with the formula V = π ∫ r² dy, substituting r = √(y²/3). Thus, the integral to calculate the volume becomes:
V = π ∫ (y²/3) dy from y=0 to y=5.
The solution to this integral gives the volume of the solid of revolution.