Which of the following integrals computes the volume of the solid formed by revolving the region bounded by the graph of y = Ln(x), the x-axis, and the line x = 3 about the line x = 3?
To find the integral that computes the volume of the solid formed by revolving the region bounded by the graph of y = Ln(x), the x-axis, and the line x = 3 about the line x = 3, we need to use the method of cylindrical shells.
First, let's visualize the region and the solid formed. The graph of y = Ln(x) looks like an increasing curve that passes through the point (1, 0) and approaches positive infinity as x approaches infinity. We want to revolve the region bounded by this curve, the x-axis, and the vertical line x = 3 around the vertical line x = 3.
To set up the integral using cylindrical shells, we need to consider an infinitesimally small strip (or shell) of height y and thickness dx along the x-axis. As we revolve this shell around the line x = 3, it forms a cylindrical shell with a radius of (3 - x) and a height of y.
The volume of this cylindrical shell can be approximated as the product of its height, thickness, and circumference:
dV = 2π(3 - x) * y * dx
Since y = Ln(x), we can write:
dV = 2π(3 - x) * Ln(x) * dx
Now, to find the total volume of the solid, we need to integrate this expression over the appropriate range of x-values. We need to consider the range of x-values that defines the region bounded by the graph of y = Ln(x), the x-axis, and the line x = 3. Looking at the graph, we can see that the region starts at x = 1 and ends at x = 3 (the vertical line x = 3).
Thus, the integral that computes the volume of the solid is:
V = ∫[1, 3] 2π(3 - x) * Ln(x) dx
To solve this integral, you can use any standard methods of integration, such as integration by parts or substitution.