9 FURTHER APPLICATIONS OF INTEGRATION
9 FURTHER APPLICATIONS OF INTEGRATION
9 FURTHER APPLICATIONS OF INTEGRATION
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GROUP WORK 1, SECTION 9.2<br />
Gabriel’s Horn<br />
Sketch the graph of f (x) = 1/x from x = 1tox =∞. (Okay, you don’t actually have to graph it all the<br />
way out to infinity.)<br />
Now think of the object formed by rotating this curve about the x-axis. This object is called Gabriel’s Horn.<br />
Your job is to paint Gabriel’s Horn. You are given a brush and as many cans of paint as you want. But before<br />
you start, it occurs to you that you might be wasting your time. After all, the horn is unbounded. You don’t<br />
want to spend hours and hours on a job that might never be finished!<br />
Here’s an idea: calculate the volume of Gabriel’s Horn. That is, find the volume for the solid obtained by<br />
rotating the region bounded by f (x) = 1/x, x = 1, and y = 0 about the x-axis. If this volume is finite, you<br />
could fill the horn with paint, shake it around a little, pour out the extra and be done with the job. (You are<br />
allowed to waste paint in Heaven.)<br />
So, can you paint the horn?<br />
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