Final answer:
To find the volume of the solid generated by revolving the region between y = x, y = 0, and x = 6 around a line, one must integrate the area of disks formed along the axis of revolution. The volume is obtained by integrating πx² from 0 to 6.
Step-by-step explanation:
When you revolve the region bounded by the graphs of the equations y = x, y = 0, and x = 6 around a given line, you create a solid of revolution. The volume of this solid can be found using calculus, specifically the method of disks or washers.
To calculate the volume, you would typically integrate the area of cross-sections perpendicular to the axis of revolution. In this case, the area of a cross-section is a disk with radius determined by the function y=x, which varies with x. Since the revolution is around the y-axis (assuming from the given equations), the volume V of the solid generated is given by integrating the squared function (since the area of a circle is πr²) from x = 0 to x = 6.
The general formula for the volume of a cylinder is V = Ah, where A is the cross-sectional area and h is the height. Applying this to our situation, the volume of the solid of revolution would be the integral of πy² from x=0 to x=6, since y=x in this region. Hence, for this particular case, the volume is calculated as π times the integral of x² from 0 to 6.