Final answer:
Using the cylindrical shells method, the volume generated by rotating the region bounded by y = 16x⁴, y = 0, and x = 1 about x = 3 is found by integrating the volume of each cylindrical shell, 2π(3 - x)(16x⁴) dx, from x = 0 to x = 1.
Step-by-step explanation:
To find the volume V generated by rotating the region bounded by the curves y = 16x4, y = 0, x = 1 about x = 3 using the method of cylindrical shells, we consider a typical cylindrical shell at a distance x from the y-axis, with thickness dx, height given by the function y = 16x4, and a radius of (3 - x).
The volume dV of this shell is given by its circumference 2π(3 - x) multiplied by its height y and thickness dx: dV = 2π(3 - x)y dx. Substituting y = 16x4 into this expression, we have dV = 2π(3 - x)(16x4) dx. To find the total volume, we then integrate dV from x = 0 to x = 1:
∫ V = ∫ 2π(3 - x)(16x4) dx from x = 0 to x = 1.
We can compute this integration by applying the properties of integrals. After finding the antiderivative and evaluating the limits, we will arrive at the solution for the volume V.