Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.x = 1 + (y − 6)2,  x = 10V =

The method of cylindrical shells is a method for finding the volume of a solid of revolution. The idea is to approximate the volume of the solid by a sum of the volumes of cylindrical shells, and then to take the limit as the number of shells goes to infinity.

Here are the steps to solve the problem:

1. Identify the region: The region is bounded by the curves x = 1 + (y - 6)^2 and x = 10.

2. Set up the integral: The volume V of the solid obtained by rotating the region about the x-axis is given by the integral ∫[a,b] 2πrh dx, where r is the radius of the cylindrical shell and h is the height of the cylindrical shell. In this case, the radius r is y and the height h is (10 - (1 + (y - 6)^2)). The limits of integration [a,b] are the y-values where the curves intersect, which are y = 3 and y = 9.

3. Evaluate the integral:

V = ∫[3,9] 2πy(10 - (1 + (y - 6)^2)) dy = 2π ∫[3,9] (10y - y + y^3 - 12y^2 + 36y - 36) dy = 2π [5y^2 - (1/2)y^2 + (1/4)y^4 - 4y^3 + 18y^2 - 36y] evaluated from 3 to 9 = 2π [(581 - (1/2)81 + (1/4)6561 - 4729 + 1881 - 369) - (59 - (1/2)9 + (1/4)729 - 427 + 189 - 363)] = 2π [3240 - 40.5 + 1640.25 - 2916 + 1458 - 324 - 45 + 4.5 - 182.25 + 108 - 54] = 2π [2053.5]

So, the volume V of the solid obtained by rotating the region bounded by the curves about the x-axis is approximately 2π * 2053.5 cubic units.