Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.

27y = x3, y = 0, x = 6

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To find the volume generated by rotating the region bounded by the curves about the x-axis using the method of cylindrical shells, we can follow these steps:

1. First, sketch the region bounded by the given curves.

The given curves are 27y = x^3, y = 0, and x = 6.

Start by drawing the x and y axes, and plot the points where y = 0 and x = 6. Then, plot the curve 27y = x^3.

2. Determine the limits of integration.

Since we are rotating the region about the x-axis, the height of the cylindrical shells will be the difference in y-values between the upper and lower curves. In this case, the lower curve is y = 0 and the upper curve is 27y = x^3.

To find the limits of integration for y, we need to solve the equation 27y = x^3 for y, and then find the y-values where x = 0 and x = 6.

When x = 0:
27y = 0^3
27y = 0
y = 0

When x = 6:
27y = 6^3
27y = 216
y = 8

So, the limits of integration for y are 0 and 8.

3. Setup the integral for the volume using the cylindrical shells method.

The volume of each cylindrical shell can be found using the formula:
dV = 2πrh * dh

Where r is the distance from the axis of rotation (in this case, the x-axis), and h is the height of the shell.

Since we are rotating about the x-axis, the distance from the axis of rotation will be x, and the height of the shell will be the difference in y-values between the curves.

So, the integral for the volume can be set up as follows:
V = ∫[0 to 8] 2π(x) * (27y - 0) dy

4. Evaluate the integral.

Integrate the equation V = 2π ∫[0 to 8] (27xy) dy.

This can be broken down into two separate integrals:

V = 2π ∫[0 to 8] (27xy) dy
= 2π(27x) ∫[0 to 8] y dy
= 2π(27x) [y^2/2] [0 to 8]

Substituting the limits of integration:
V = 2π(27x) [(8^2/2) - (0^2/2)]
= 2π(27x) (32)
= 1728πx

5. Substitute the value of x.

Since we want to find the volume for the entire region, we need to evaluate V for x = 6 (the farthest point).

V = 1728π(6)
= 10368π

Therefore, the volume generated by rotating the region bounded by the given curves about the x-axis is 10368π cubic units.