3. (Gabriel’s horn) Consider the surface obtained by revolving about the x-axis

Question

3. (Gabriel’s horn) Consider the surface obtained by revolving about the x-axis the curve y = 1/x, x 1.

(a) Express the volume (of revolution) of the solid inside this surface as an improper integral and evaluate this integral.

(b) Express the area of the surface as an improper integral and use the comparison test 1

to show that the integral diverges. (Hint: For x1, sqrt(1+x^4) sqrt(1+0)=1.)

(c) The above says that the volume of the horn is finite, but its surface area is infinite. In practical terms this seems to indicate that the horn can be filled with a finite amount of paint, but there would not be enough to paint its inside surface. Explain what the flaw in this apparent contradiction is. Write down your explanation in complete sentences.

Explanation / Answer

3. (Gabriel’s horn) Consider the surface obtained by revolving about the x-axis

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