Final answer:
To find the volume of the solid of revolution generated by revolving the region bounded by the x-axis and the graph of y=x√2-x about the x-axis, the method of cylindrical shells can be used. The volume of each shell is calculated as the product of its circumference, height, and thickness. Integrating the volumes of all the shells gives the total volume of the solid.
Step-by-step explanation:
To find the volume of the solid of revolution generated by revolving the region bounded by the x-axis and the graph of y=x√2-x about the x-axis, we can use the method of cylindrical shells.
The volume of each shell can be calculated as the product of the circumference of the shell, the height of the shell (which is the difference in y-values between the curve and the x-axis), and the thickness of the shell (which is an infinitesimally small change in x).
Integrating the volumes of all the shells from the lower limit of x=0 to the upper limit of x=b (where b is the x-coordinate where the curve intersects the x-axis), we can find the total volume of the solid.
Therefore, the volume of the solid of revolution is given by the integral ∫[0,b] 2πx(y) dx, where y(x) = x√2-x.