Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

Impossible Integral?

The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives:

INTEGRAL0 to infinity exp(-x2) dx.

Why? Well, there isn't a closed-form expression for the antiderivative of the integrand, so the Fundamental Theorem of Calculus won't help. But the expression is meaningful, since the it represents the area under the curve from 0 to infinity.

Furthermore, there is a nice trick to find the answer! Call the integral I. Multiply the integral by itself: this gives

I2 = [ INTEGRAL0 to infinity exp(-x2) dx ] [ INTEGRAL0 to infinity exp(-y2) dy ]

then view as an integral over the first quadrant in the plane:

= [ INTEGRAL0 to infinity INTEGRAL0 to infinity exp(-x2-y2) dx dy]

then change to polar coordinates (!):

= INTEGRAL0 to Pi/2 INTEGRAL0 to infinity exp(-r2) r dr d(THETA).

Now this is quite easy to evaluate: you find that I2=Pi/4. This means that I, the original value of the integral you were looking for, is Sqrt[Pi]/2.


Presentation Suggestions:
This trick is often learned in multivariable calculus course; it is best to show it right after learning to integrate in polar coordinates. If polar coordinates have not been introduced yet, you can view the squared integral as the volume of a solid of revolution, and evaluate using shells.

The Math Behind the Fact:
You may recognize the integrand as the familiar (unscaled) bell curve. An alternate way of evaluating this integral (without appealing to an unmotivated trick!) is to view it as a complex integral and use residue theory. You can learn more about this in a course on complex analysis.

How to Cite this Page:
Su, Francis E., et al. "Impossible Integral?." Math Fun Facts. <>.

Keywords:    multivariable calculus
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!