Consider the region below the graph of y = 1/x for x ge 1. This problem is often
ID: 2868635 • Letter: C
Question
Consider the region below the graph of y = 1/x for x ge 1. This problem is often referred to as the "Painter's Paradox". Why do you think that is? (Consider trying to paint the outside of this solid and/or filling the solid with paint.) Geogebra Applet Instructions Goal: Create a Geogebra applet that gives the area under f(x) = 1/x (or any function) from 1 to R. Type the function f(x) = 1/x into the input bar. Since R will vary, let's create a slider. Type in the input bar "R=2". Now right click on the "R=2" in the Algebra window, and choose "Show Object" (or select the bubble next to it). In the input bar, type integral[f, 1, R]. The new value "a" should appear under Dependent Objects. Its value is the area under f(x)on [1, R]. Try moving the slider for R to see how the area changes. You can change the bounds and increments of R by right clicking, and going to Object Properties. By looking at very large values of R, it can help in determining if f/(x)dx converges or diverges, and that's what we're going to do in this lab. You can change the function, f(x), to something else by simply double-clicking. In this lab, we'll want to change this function into whatever function (area, volume, surface area, etc.) we're interested in integrating. Change the rounding to 5 decimal places by going to Options rightarrow Rounding.Explanation / Answer
Gabriel's Horn (also called Torricelli's trumpet) is a geometric figure, which has infinite surface area but finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinitely large section of the x-y plane about the x-axis generates an object of finite volume was considered paradoxical.
Actually, while the section lying in the x-y plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the 'weighted sum' of sections, is finite.
Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its inner surface