Final answer:
To find the volume of the solid generated by revolving the region bounded by y=3x, y=0, and x=4, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by y=3x, y=0, and x=4, we can use the method of cylindrical shells.
The region bounded by y=3x, y=0, and x=4 forms a right triangle with base 4 and height 12.
The formula for the volume of the solid generated by revolving a region around the x-axis using cylindrical shells is V = 2π ∫(radius)(height) dx, where the radius is the distance from the axis of rotation to the curve and the height is the infinitesimal width of the shell.
In this case, the radius is x and the height is 3x.
So, the volume of the solid is V = 2π ∫(x)(3x) dx = 2π ∫(3x^2) dx = 2π[(x^3)/3] evaluated from 0 to 4.
Plugging in the limits of integration, we get V = 2π((4^3)/3 - (0^3)/3) = 128π/3.
Therefore, the volume of the solid generated by revolving the region bounded by y=3x, y=0, and x=4 is 128π/3 cubic units.