If the vertical line x = 2 is the axis of rotation, and the area is the area bounded by the curve (which is a sideways parabola opening to the right with a vertex at (1, 5)), and we are finding the volume by shells, then a representative strip will be vertical (parallel to the axis of rotation), of width dx. Thus, we need to integrate with respect to x. Our bounds of integration are then x = 1 and x = 2.
A representative shell will have a surface area = 2πrh, which means we need the radius and height in terms of x.
The radius is (2-x). The height is the vertical height between the two branches of the parabola: y = 5 +√(x-1) and y = 5 - √(x-1). So the height is 2√(x-1).
V = ∫12 (2π)(2 - x)(2√(x-1))dx ...
