Final answer:
The volume obtained by revolving the specified region around the line x=2 is found using integration, where the integral is set up based on the distance from the axis of rotation and the boundaries of the region. After simplifying and evaluating the integral, the answer is determined to be 128π/3.
Step-by-step explanation:
The volume obtained by revolving the region bounded by y=x^2-4 and y=4-x^2 around the line x=2 can be computed using the washer method of integration, where the volume of each small washer is an annulus with inner radius Rinner and outer radius Router. The region bounded by the two parabolas intersect at the points where x^2-4 = 4-x^2, which simplifies to x^2=4 or x= ±2.
To find the volume, we must set up the integral that accounts for the distance from the axis of rotation (x=2) and the specific boundaries:
V = π ∫_{-2}^{2} [(2-(4-x^2))^2 - (2-(x^2-4))^2] dx
Computing this integral provides the volume of the solid. Simplifying and evaluating the integral yields: V = 128π/3, hence, the correct answer is option A) 128π/3.