Question

Shown above is the graph of y=f(x)=1/x, Consider the infinitely long solid pictured below, formed by rotating the region defined by x≥4 and 0≤y

Answer 1

The question pertains to a hypothetical solid generated by rotating the graph of f(x) = 1/x around the x-axis starting at x=4. The characteristics of the function, including its asymptotic behavior and the method to calculate the volume of the solid through integration, are integral to understanding the problem.

Step-by-step explanation:

The question revolves around the function f(x) = 1/x and involves understanding the behavior of this function when x gets very large or very small. The function f(x) = 1/x has asymptotes, which are lines that the graph approaches but never actually touches. Since the function is defined only for x≠0, it means that as x approaches zero, y approaches infinity, a behavior typical of a hyperbolic function.

When the given solid is formed by rotating the region defined by x≥4 and 0≤y, you're actually dealing with the concept of the volume of revolution. Such volumes are typically calculated using integrals, and in this case, one would set up an integral from x=4 to infinity of the function pi*(1/x)² dx to calculate the volume.

The steps to plot the function between x=0 to x=20 would be as follows:

1. Label the graph with f(x) and x.
2. Scale the x-axis from 0 to 20.
3. Plot the value of the function f(x) at various points within this interval.
4. Connect the points to form the graph of the function.