Final answer:
To calculate the volume of the solid formed by revolving the region between y=4x-x² and y=0 around the y-axis, one must perform an integral using the shell method for cylindrical shells with radius x and height 4x - x² from x=0 to x=4.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the curves y = 4x - x² and y = 0 about the y-axis, we can use the shell method. This method involves integrating the volume of cylindrical shells. The boundaries for x are determined by the intersections of the curve with the x-axis, which happens when y=0. Solving 4x - x² = 0, we can find the x-intercepts to be 0 and 4.
The formula for the volume of a cylindrical shell with radius r and height h is 2πrh × (dr), where r is the average radius of the shell, h is the height, and dr is the thickness of the shell. In our case, h = 4x - x² and r = x. Integrating with respect to x from 0 to 4, we can find the volume of the entire solid by the shell method as follows:
V = ∫ 2πx(4x - x²)dx from x = 0 to x = 4.
When we perform this integration, the result is the volume of the solid of revolution which is V = (8π/3) × 16, after simplifying the integral.