Final answer:
The volume of the solid created by rotating the region bounded by y=√x and y=x about the line x=6 is determined using the washer method which involves calculating the volume of a thin washer and integrating this across the range of intersection points of the two curves.
Step-by-step explanation:
The question involves finding the volume of a solid obtained by rotating a region bounded by the curves y=√x and y=x about the line x=6. To do this, we use the disk method or the washer method, which involves calculating the volume of an infinitesimally thin washer and integrating this across the range of x values where the two curves intersect.
We must first identify the radius of each disk/washer. Since the solid is rotated about the line x=6, the outer radius (R) for the washer is 6 - y=x and the inner radius (r) is 6 - y=√x. Then, we calculate the area of one such washer, which is given by π(R^2 - r^2), and integrate this across the intersection points of the two curves to find the total volume.
To find the exact intersection points, one needs to set √x equal to x and solve for x. Then, with the established limits of integration, the integral becomes ∫ π[(6-x)^2 - (6-√x)^2] dx from the first intersection point to the second. This integral, once solved, will give the volume of the solid.