Theory and ExamplesThe infinite paint can or Gabriel’s

Chapter 8, Problem 76E

(choose chapter or problem)

Problem 76E

Theory and Examples

The infinite paint can or Gabriel’s horn As Example 3 shows, the integral  diverges. This means that the integral

which measures the surface area of the solid of revolution traced out by revolving the curve y = 1/x, 1 ≤ x, about the x-axis, diverges also. By comparing the two integrals, we see that, for every finite value b>1.

However, the integral

 for the volume of the solid converges.

a. Calculate it.

b. This solid of revolution is sometimes described as a can that does not hold enough paint to cover its own interior. Think about that for a moment. It is common sense that a finite amount of paint cannot cover an infinite surface. But if we fill the horn with paint (a finite amount), then we will have covered an infinite surface. Explain the apparent contradiction.