Final answer:
To calculate the volume of a solid generated by revolving a region bounded by curves around the y-axis, the disk or washer methods are used, involving integration of the volume formula for disks or washers across the x-interval.
Step-by-step explanation:
To find the volume of the solid generated when a region bounded by curves is revolved about the y-axis, we can use the disk or washer method, depending on the region and the axis of revolution. When the region is simply the area under a curve from one x-value to another, the disk method involves slicing the solid into thin disks perpendicular to the axis of revolution. The volume of each disk is πr²h, where r represents the radius of the disk and h is the height, or thickness, of the disk, which in this case becomes an infinitesimally small change in x (often written as dx).
For a region bounded between two curves, we would use the washer method, which is similar to the disk method but subtracts the volume of the inner disk from the outer disk. This is necessary when there is a 'hole' in the region being revolved. Thus, the volume of a washer is π(R² - r²)h, where R and r represent the outer and inner radii, respectively, and h here stands for the thickness of the washer (again, an infinitesimal dx).
In both methods, the total volume is found by integrating the volume formula for the disks or washers across the interval of interest, which gives us the sum of the volumes of all infinitesimally thin disks or washers that make up the solid.