Find the volume of the solid generated by revolving the region bounded by y = 8, x= 2, and x =7 about the x-axis. The region rotated about the x-axis to generate the solid is the shaded region in the figure. 10-7 A cross-section of the solid is a disk perpendicular to the x-axis with radius the segment from the x-axis to the line y = 8. The volume, then, is V= | n (f(x))2dx. The volume expressed as an integral in terms of x only is V = | x(8)~dx. Now find the antiderivative of the integral. V= =(8) dx = [64xx]2 2 Evaluate the definite integral. [64xx], = 320x Therefore, the volume is 320x cubic units