Final answer:
The volume of the solid formed by rotating the region bounded by f(x) and the x-axis on [0, b] around the y-axis is calculated using the disk method, integrating πf(x)^2 from 0 to b.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the function f(x) and the x-axis on the interval [0, b] around the y-axis, we can use the disk method. The volume of a thin disk with radius r and thickness dx is dV = πr^2dx. By integrating this expression from 0 to b, we obtain V = π ∫_{0}^{b} f(x)^2 dx. This calculation step produces the total volume when the defined region is revolved about the y-axis.To find the volume of the solid generated when the given region is revolved about the y-axis, we can use the method of cylindrical shells. First, let's express the function f(x) in terms of y so we can integrate with respect to y. Next, we'll find the limits of integration for y by setting f(x) = 0 and solving for x. Then, we'll set up the integral to calculate the volume of each cylindrical shell. Finally, we'll integrate the volume function over the interval [0, b] to find the total volume.
It is important to understand the concept rather than copying from existing sources. In this case, one must identify that the radius of the disk in terms of the function is r = f(x) and the thickness of the disk is an infinitesimal increment along the x-axis dx. The resulting integration will provide the desired volume.