Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.y = 27x3,  y = 0,  x = 1;  about x = 2

To find the volume of the solid obtained by rotating the region bounded by the curves y = 27x^3, y = 0, x = 1 about the line x = 2, we can use the method of cylindrical shells.

The formula for the volume of a solid of revolution using cylindrical shells is V = 2π ∫ from a to b [f(x) * (x - c)] dx, where f(x) is the function being revolved, c is the line of revolution, and [a, b] is the interval over which we're integrating.

Here, f(x) = 27x^3, c = 2 (the line of revolution), and the interval of integration is [0, 1] (from x = 0 to x = 1).

So, we have:

V = 2π ∫ from 0 to 1 [27x^3 * (2 - x)] dx

To solve this integral, we can distribute the x inside the brackets:

V = 2π ∫ from 0 to 1 [54x^3 - 27x^4] dx

Now, we can integrate term by term:

V = 2π [ (54/4)x^4 - (27/5)x^5 ] from 0 to 1

Evaluating this from 0 to 1 gives:

V = 2π [(54/4) - (27/5)] - 0

Simplify the fractions:

V = 2π [(13.5) - (5.4)]

V = 2π [8.1]

So, the volume V of the solid is approximately 50.27 cubic units.