Final answer:
To find the volume of the solid generated by revolving a region about the x-axis using the shell method, we need to integrate the circumference of each shell multiplied by its height.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the equation x = 2y - y^2 about the x-axis using the shell method, we need to integrate the circumference of each shell multiplied by its height.
The equation x = 2y - y^2 represents a parabola opening downwards. To find the limits of integration, we set the equation equal to zero and solve for y. We will obtain two values, which will be our lower and upper limits of integration.
Next, we find the circumference of each shell by using the formula C = 2πrh, where r is the radius of each shell and h is the height. The radius can be expressed as x - y, and the height can be expressed as the difference between the upper and lower functions.
Integrating the circumference multiplied by the height over the limits of integration will give us the volume of the solid.