Final answer:
The integral to determine the volume of the solid generated by revolving the region bounded by the curves y = x² - 4x and y = 4x about the line y = -5 is V = 2π ∫_{0}^{8} (x² - 8x + 5)(8x - x²) dx.
Step-by-step explanation:
To calculate the volume of the solid generated by revolving the region bounded by the curves y = x² - 4x and y = 4x about the line y = -5, we use the method of cylindrical shells. The volume V of the solid can be found by integrating the difference in the areas of the two functions rotated around the line y = -5.
First, we find the intersection points of the curves by setting them equal to each other:
- x² - 4x = 4x
- x² - 8x = 0
- x(x - 8) = 0
From this, we find the intersection points at x = 0 and x = 8. As we revolve around the line y = -5, the radius of the shell at any point x is R(x) = x² - 4x + 5 (from the top function minus the bottom function plus the distance to y = -5).
The height of the cylindrical shell is the difference between the functions, which is h(x) = 4x - (x² - 4x) = 8x - x², and the volume of a cylindrical shell is given by V = 2π ∫ from a to b of (R(x) ⋅ h(x) dx).
Therefore, the integral for calculating the volume V is:
V = 2π ∫_{0}^{8} (x² - 8x + 5)(8x - x²) dx.