Final Answer:
The volumes of the solids generated by revolving the regions bounded by the graphs of the equations y = x² and y = 6x − x² about the x-axis can be found by applying the disk method. The final answer is 560/3 π cubic units.
Step-by-step explanation:
To find the volumes of the solids, we use the disk method, which involves integrating the cross-sectional area of the solid perpendicular to the axis of rotation. The region bounded by the graphs of y = x² and y = 6x − x² is the area between the curves. The points of intersection can be found by setting x² equal to 6x − x² and solving for x, resulting in x = 2 and x = 3.
The differential area A of each disk at a given x is π ⋅ [f(x)]², where f(x) is the distance between the curves at x. In this case, f(x) = (6x - x²) - x² = 6x - 2x². The integral for the volume is then:
V=∫23π⋅[(6x−2x2)2−x4]dx
Solving this integral results in 560/3 π cubic units.
In summary, the disk method is employed to find the volumes of solids generated by revolving the region between the given curves about the x-axis. The integral is set up with the appropriate differential area formula, and after evaluation, the final answer is 560/3 π cubic units.