Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 1 sec x and y = 3 about the line y = 1, we can use the method of cylindrical shells. This involves integrating the circumference of each shell multiplied by its height and thickness.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 1 sec x and y = 3 about the line y = 1, we can use the method of cylindrical shells. This involves integrating the circumference of each shell multiplied by its height and thickness. The height of each shell is given by the difference between the y-values of the curves at a given x-value, and the thickness is infinitesimally small.
The equations for the curves are y₁ = 1 sec(x) and y₂ = 3. To find the limits of integration, we need to find the x-values at which the curves intersect. Setting y₁ = y₂, we have 1 sec(x) = 3. Solving for x, we get x = arccos(1/3).
The volume of the solid can then be found by integrating from x = 0 to x = arccos(1/3) the following expression: (2πy₁)(y₂ - y₁)dx.