Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 3 sec(x) and y = 5 about the line y = 3 sec(x), we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 3 sec(x) and y = 5 about the line y = 3 sec(x), we can use the method of cylindrical shells.
First, we need to find the limits of integration. The curves intersect at two points: x = 0 and x = π/3. Thus, the limits of integration will be from 0 to π/3.
Next, we need to find the height of each cylindrical shell. The height will be the difference between the upper curve and the lower curve at each value of x. In this case, the height is (5 - 3sec(x)).
Finally, we integrate the volume of each cylindrical shell using the formula V = 2πrh, where r is the distance from the axis of rotation to the curve at each value of x, and h is the height. After integrating, we obtain the volume of the solid.