Final answer:
Calculate the volume of the solid by integrating the area of circular disks from x=0 to x=1.5, using the curve y=6-4x as the radius and revolving around the x-axis.
Step-by-step explanation:
To find the volume of the solid generated by revolving the shaded region about the x-axis, we can use the method of disks or washers. The region is bounded by the curves y = 6 - 4x, y = 0, and x = 0. The function y = 6 - 4x represents the radius of each disk as a function of x, and the volume of a disk with thickness dx would be π × radius² × dx. To find the total volume, we integrate this expression from the bounds of x=0 to the x-intercept of the line y = 6 - 4x, which is at x = 1.5 (since 6 - 4(1.5) = 0).
Hence, the integral to compute the volume V is:
V = ∫01.5 π(6 - 4x)² dx
By simplifying and computing the integral, we would obtain the total volume of the solid of revolution.