Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 4x and y = 4x about the line y = 4, 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 y = 4x and y = 4x about the line y = 4, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by 2πrhΔx, where r is the distance from the shell to the axis of rotation, h is the height of the shell, and Δx is the width of the shell.
In this case, the curves y = 4x and y = 4x intersect at (0, 0) and (1, 4). So, the range of x values for the region of interest is from 0 to 1. The distance from the shell to the axis of rotation is 4x (since y = 4x), the height of the shell is 4 (since y = 4), and the width of the shell is Δx.
Therefore, the volume of the solid can be found by integrating the expression 2π(4x)(4)Δx from x = 0 to x = 1:
V = ∫01 2π(4x)(4)Δx = ∫01 32πxΔx