Final answer:
The volume of a solid obtained by rotating a region bounded by the curves x = 3(y - 4)² and x = 4 about the x-axis using cylindrical shells involves integrating the product of the radius, height, and circumference of the shells. The formula for the volume of a sphere is 4/3 π r³, which is distinct from its surface area formula 4 π r².
Step-by-step explanation:
The student's question involves finding the volume of a solid obtained by rotating a region about the x-axis using the method of cylindrical shells. The region is bounded by the curves x = 3(y - 4)² and x = 4. To solve this, we would integrate the function representing the radius of each cylindrical shell times its height and 2π (the circumference of the circle at the end of the shell) with respect to y, because the shells' axis of rotation is the x-axis.
It's worth mentioning that the question at hand evokes the concept of a sphere's volume because cylindrical shells can be used to derive such volumes. The volume of a sphere is given by the formula volume = 4/3 π r³. This is in contrast to a formula for the surface area, 4 π r². For a cylinder, the volume formula is V = Ah where A is the area of the base and h is the height.
To answer the provided question about the volume of a sphere, the correct expression is 4³/3. When considering volumes and surface areas, it is important to remember the distinction between linear, squared, and cubed dimensions in geometric formulae.