Final answer:
The integral that will determine the volume is ∫[0,4] (π(x²-3x+4)² + π(4x+4)²) dx. This formula represents the integration over the interval [0,4]. So, the option 4 is correct.
Step-by-step explanation:
In order to ascertain the volume of the solid formed by revolving the region enclosed by the curves y=x²-3x and y=4x around the line y=-4, the integral expression is ∫[0,4] (π(x²-3x+4)² + π(4x+4)²) dx.
This formula represents the integration over the interval [0,4], multiplying by π, and squaring the difference between the outer radius (distance from the line y=-4 to y=4x) and the inner radius (distance from the line y=-4 to y=x²-3x).
The outer radius is determined by the curve y=4x, while the inner radius is determined by the curve y=x²-3x.
The integration captures the summation of these squared differences, providing the volume of the solid of revolution.
This mathematical approach ensures an accurate representation of the spatial extent of the generated solid.
Hence, the option 4. ∫[0,4] (π(x²-3x+4)² + π(4x+4)²) dx, is correct.