Final answer:
The volume of the solid formed by rotating the given region about the axis x = -8 can be determined using the method of cylindrical shells by integrating from the points of intersection of the curves, setting up the integral with appropriate bounds and functions, and calculating the result.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region enclosed by the curves y = x² − 3 and y = 3x − 3 about the axis x = −8, we can use the method of cylindrical shells. The volume of a single cylindrical shell with radius r and height h is given by the formula V = 2πrh. The radius of the shell, in this context, is the distance from the axis of rotation (x = −8) to x, which is r = x + 8. The height of the shell is the difference between the functions, so h = (3x − 3) - (x² − 3). Given this, we will integrate to find the total volume.
We'll first find the points of intersection to establish the limits of integration by setting y = x² − 3 = 3x − 3. Solving for x gives us the points x = 0 and x = 3. The integration will thus be performed from x = 0 to x = 3.
The integral to find the volume V is then:
V = ∫[0,3] 2π (x + 8) [(3x − 3) − (x² − 3)] dx, which simplifies and calculates the required volume.