Final answer:
To compute the volume of a solid formed by revolving a region around a line, one uses integral calculus with the disk or washer method for horizontal lines and cylindrical shells method for vertical lines, depending on the axis of revolution.
Step-by-step explanation:
To find the volume of the solid generated when a region bounded by a function and the x-axis is revolved about a line, you can use the disk or washer method if revolving around a horizontal line, and the cylindrical shells method if revolving around a vertical line. For the disk or washer method, you integrate the area of the disk or washer cross-sections perpendicular to the axis of revolution. The cylindrical shells method involves integrating the surface area of cylindrical 'shells' parallel to the axis of revolution.
The equation of the function and the line of revolution are needed to set up the integral. For example, if the function was f(x)=x^2 and the line of revolution was y=1, and the region was bounded on interval [a,b], you would calculate the integral from a to b of π[(1-f(x))^2] to find the volume of the solid using the washer method.
Different geometries like cylinders or paraboloids may result from different functions and lines of revolution. The cross-sectional area plays a critical role in determining the volume of the solid generated.