Final answer:
To find the volume of the solid of revolution, we determine the outer and inner radii using the curves y = x² −2x−2 and y = x + 2 and the line y = -3, set up the integral with limits based on the intersection points, and then calculate the volume.
Step-by-step explanation:
To find the volume of the solid of revolution created by revolving the area bounded by the curves y = x² −2x−2 and y = x + 2 about the line y = −3, we can use the method of cylindrical shells or the disk/washer method. Since the axis of rotation is not mentioned as the x-axis or y-axis, we'll assume we're using the washer method because the axis of rotation is parallel to the x-axis.
To set up the integral, we first need to find the points of intersection for the curves y = x² −2x−2 and y = x + 2. Solving for x, we get the intersection points. Using these points as limits of integration, the volume V is given by the integral
V = π ∫ ( ext{outer radius})² - ( ext{inner radius})² dx
The outer radius is the distance from the axis of rotation (y = -3) to the curve y = x + 2, and the inner radius is the distance from the axis of rotation to the curve y = x² −2x−2. Substituting these into the integral and solving, we obtain the volume of the solid.