Final answer:
The volume of the solid obtained by rotating the region bounded by the curves x=y² and x=1 about the line x=8 can be found using the method of cylindrical shells. The volume of each cylindrical shell is given by 2πrh, where r is the distance from the line x=8 to the strip and h is the height of the strip. The integral of the volume of each cylindrical shell can be evaluated to find the total volume.
Step-by-step explanation:
The volume of the solid obtained by rotating the region bounded by the curves x=y² and x=1 about the line x=8 can be found using the method of cylindrical shells.
The region bounded by the curves forms a shape that resembles a funnel. To find the volume, we need to integrate the volume of each cylindrical shell formed by rotating a small vertical strip of the region about the line x=8.
The volume of each cylindrical shell is given by 2πrh, where r is the distance from the line x=8 to the strip and h is the height of the strip.
Since the curves x=y² and x=1 intersect at point (1,1), the radius of each cylindrical shell is given by the difference between 8 and the x-coordinate of the strip.
Therefore, the radius r is 8-x. The height of the strip h is given by the difference between the y-coordinates of the curves at the x-coordinate of the strip. Therefore, the height h is y²-1.
To find the limits of integration, we need to find the values of x for which the curves x=y² and x=1 intersect. Setting y²=1, we get y=±1. Therefore, the curves intersect at points (1,1) and (1,-1).
Now, we can integrate the volume of each cylindrical shell from x=1 to x=8 using the limits of integration we found. The integral is given by ∫[1,8] 2π(8-x)(y²-1) dx.