Final answer:
To find the volumes of the solids generated by revolving the region in the first quadrant bounded by the curve x = y - y^3 and the y-axis about the given axes, we can use the method of cylindrical shells and integrate the volume formula.
Step-by-step explanation:
To find the volumes of the solids generated by revolving the region in the first quadrant bounded by the curve x = y - y^3 and the y-axis about the given axes, we need to use the method of cylindrical shells. We can divide the region into small vertical strips and imagine rotating each strip around the given axes to form a cylindrical shell. The volume of each cylindrical shell can be calculated using the formula V = 2πrhΔx, where r is the distance from the axis of rotation to the strip, h is the height of the strip, and Δx is the width of the strip. By integrating this formula over the range of y-values that define the region, we can find the total volume of the solids generated.