Final answer:
The integral for the volume of the solid created by rotating the region Q around x=1 between y=1 and y=2 is found using cylindrical shells. It is set up as V = 2π ∫_{1}^{2} (1 - u(y))(v(y) - u(y)) dy, by considering the circumference, the height of the shells, and the thickness.
Step-by-step explanation:
The student is asking for the integral that describes the volume of the solid created by rotating the region Q, defined by two functions u(y)=y√ 1 and v(y)=2, around the line x=1 for the interval between y=1 and y=2. In mathematics, this type of problem is typically solved using the method of cylindrical shells or the disk/washer method. However, the given functions suggest the use of cylindrical shells as one function is linear of y and the other is a constant.
To set up the integral, we need to consider the distance from each infinitesimally thin shell to the axis of rotation (x=1) and the height of the shell given by the difference between v(y) and u(y). The volume V of the solid is thus obtained by integrating the product of the circumference of each shell, its height, and its thickness, from y=1 to y=2.
The integral to calculate the volume V is:V = 2π ∫_{1}^{2} (1 - u(y))(v(y) - u(y)) dy
This integral can be evaluated by substituting the given functions into it and performing the integration.