Final answer:
Using the method of cylindrical shells, the volume of the solid formed by rotating the region between y = 8 - x² and y = x² around the axis x = 2 is obtained by integrating the volume of infinitesimally thin shells from 0 to √2.
Step-by-step explanation:
We are asked to find the volume of the solid formed by rotating the region between y = 8 - x² and y = x² around the axis x = 2 using the method of cylindrical shells. To apply this method, we consider a typical shell at a distance x from the y-axis with height y, and using the formula V = 2π∑(Rh), where R is the distance from the axis of rotation to the shell and h is the height of the shell.
The volume of such a shell is the product of its circumference (2πR), its height (h = 8 - 2x²), and the thickness of the shell (dx).
The limits of integration are from 0 to √2, since the intersection points of y = 8 - x² and y = x² are found by setting them equal to each other and solving for x. Thus, we can write the integral for the volume V as:
V = ∫_{0}^{√2} 2π (2 - x) (8 - 2x²) dx.
After evaluating this integral, we will obtain the volume of the solid created by the rotation of the given region around x = 2.