Final answer:
To find the area of the region satisfying the given inequalities, find the intersection of the shaded regions formed by each inequality. To find the volume of the solid generated by revolving the region about the y-axis, use the method of cylindrical shells. To find the volume of the solid generated by revolving the region about the x-axis, use the method of disks or washers.
Step-by-step explanation:
To find the area of the region satisfying the given inequalities, we need to find the intersection of the shaded regions formed by each inequality. This intersection will give us the region of interest. Once we have the region, we can use standard geometric formulas to find its area.
To find the volume of the solid generated by revolving the region about the y-axis, we can use the method of cylindrical shells. We divide the region into infinitely many cylindrical shells and integrate their volumes to find the total volume of the solid.
To find the volume of the solid generated by revolving the region about the x-axis, we can use the method of disks or washers. We divide the region into infinitely many disks or washers and integrate their volumes to find the total volume of the solid.