Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4.

y = 3 x^4 y = 0 x = 2

To find the volume generated by rotating the region bounded by the curves y = 3x^4, y = 0, and x = 2 about the line x = 4, we can use the method of cylindrical shells.

The first step is to graph the curves and identify the region bounded by them. In this case, we have a curve y = 3x^4, the x-axis (y = 0), and the vertical line x = 2. The region bounded by these curves is a solid shape between the x-axis and the curve from x = 0 to x = 2.

Next, we need to consider an infinitesimally thin strip within this region, which will form a cylindrical shell when rotated about the line x = 4. The height of this strip will be the difference between the y-values of the curves, which is given by y = 3x^4 - 0 = 3x^4.

To calculate the radius of the cylindrical shell, we consider the distance from the line of rotation, x = 4, to the curve. Since we are rotating about a vertical line, the radius will be the x-coordinate of each point in the region. Thus, the radius is given by r = 4 - x.

The thickness of the cylindrical shell is an infinitesimally small change in x, which we can represent as Δx. The volume of each cylindrical shell can be calculated as the product of its height, radius, and thickness, which gives us dV = 2π(4 - x)(3x^4)Δx.

To find the total volume, we integrate this expression from x = 0 to x = 2:

V = ∫[0,2] 2π(4 - x)(3x^4) dx

Now, you can find the antiderivative of the integrand and evaluate the definite integral to calculate the volume.