Here’s a simple Calc 1 problem:
Before you read any of my own commentary, what do you think? Does this integral converge or diverge?
image from illuminations.nctm.org
Many textbooks would say that it diverges, and I claim this is true as well. But where’s the error in this work?
Did you catch any shady math? Here’s another equally wrong way of doing it:
This isn’t any more shady than the last example. The change in the bottom limit of integration in the second piece of the integral from a to 2a is not a problem, since 2a approaches zero if a does. So why do we get two values that disagree? (In fact, we could concoct an example that evaluates to ANY number you like.)
Okay, finally, here’s the “correct” work:
But notice that we can’t actually resolve this last expression, since the first limit is and the second is and the overall expression has the indeterminate form . In our very first approach, we assumed the limit variables and were the same. In the second approach, we let . But one assumption isn’t necessarily better than another. So we claim the integral diverges.
All that being said, we still intuitively feel like this integral should have the value 0 rather than something else like . For goodness sake, it’s symmetric about the origin!
In fact, that intuition is formalized by Cauchy in what is called the “Cauchy Principal Value,” which for this integral, is 0. [my above example is stolen from this wikipedia article as well]
I’ve been debating about this with my math teacher colleague, Matt Davis, and I’m not sure we’ve come to a satisfying conclusion. Here’s an example we were considering:
If you were to color in under the infinite graph of between -1 and 1, and then throw darts at the graph uniformly, wouldn’t you bet on there being an equal number of darts to the left and right of the y-axis?
Don’t you feel that way too?
(Now there might be another post entirely about measure-theoretic probability!)
What do you think? Anyone want to weigh in? And what should we tell high school students?
**For a more in depth treatment of the problem, including a discussion of the construction of Reimann sums, visit this nice thread on physicsforums.com.