Final answer:
To find the volume generated by rotating a region about a line, use the method of cylindrical shells. Set up the volume formula using the radius, height, and width of the shells, then integrate over the range of x-values defining the region.
Step-by-step explanation:
To find the volume generated by rotating a region about a line, we can use the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = 2πrhΔx, where r is the distance from the line of rotation to the shell, h is the height of the shell, and Δx represents the width of the shell. We need to integrate this formula over the range of x-values that defines the region.
First, we need to determine the limits of integration. These are the x-values between which the region lies. To find these limits, we can set the equations of the curves that bound the region equal to each other and solve for x. The resulting values of x will give us the limits of integration.
Once we have the limits of integration, we can then calculate the volume by integrating the formula V = 2πrhΔx with respect to x over the specified range.