Introduction: “One of the more beautiful results related to approximating $\pi $ is the integral

Since the integrand is nonnegative on the interval $[0,1]$, this shows that $\pi $ is strictly less than $22/7$, the well known approximation to $\pi $. The first published statement of this result was in 1971 by *D. P. Dalzell* [Eureka 34, 10–13 (1971)], although anecdotal evidence [see J. M. Borwein, The life of Pi, history and computation, seminar presentation 2003, available from

An obvious question at this point might be whether similar elegant integral results can be found for other rational approximations for $\pi $. A particularly good approximation is 355/113, which is accurate to seven digits. Our aim here is to find a variety of such integral results.”

However, despite several variations of the style of integrand, no simple and elegant result was found.

The article is highly recommended as a basis of an undergraduate project.

##### MSC:

11Y60 | Evaluation of constants |