Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves x = 4/3y, x = 0, y = 3 about the y-axis, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves, we can use the method of cylindrical shells. The volume of a cylindrical shell is given by 2πrhΔy, where r is the distance from the y-axis to the shell, h is the height of the shell, and Δy is the thickness of the shell.
In this case, the distance from the y-axis to the curve x = 4/3y is y/3, and the height of the shell can be calculated as the difference between the x-values of the curves x = 4/3y and x = 0, which is (4/3y - 0) = 4/3y. Therefore, the volume of each shell is 2π(y/3)(4/3y)Δy.
To find the total volume, we need to integrate this expression from y = 0 to y = 3. The integral of 2π(y/3)(4/3y) with respect to y is equal to πy^3.