Final answer:
The integral ∫[0,4] (π(3x+4)² - π(x²-4x+4)²) dx will determine the volume of the solid generated by revolving the given region about the line y=-4.
Step-by-step explanation:
The volume of the solid that is generated by revolving the region bounded by the curves y=x²-4x and y=3x about the line y=-4 can be determined using the integral:
∫[0,4] (π(3x+4)² - π(x²-4x+4)²) dx
This integral represents the difference between the volumes of the two solids formed by revolving the curves about the given axis. The first term, π(3x+4)², corresponds to the volume of the solid formed by revolving the curve y=3x, while the second term, π(x²-4x+4)², represents the volume of the solid formed by revolving the curve y=x²-4x.