Introduction: “One of the more beautiful results related to approximating $\pi$ is the integral $$\int\sb 0\sp 1\frac{x\sp 4(1-x)\sp 4}{1+x\sp 2}\,dx=\frac{22}{7}-\pi. \tag1$$ 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 {\it 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 \url{http://www.cecm.sfu.ca/ jborwein/pi-slides.pdf}, March 2005] suggests it was known by Kurt Mahler in the mid-1960s. The result (1) is not hard to prove, if perhaps somewhat tedious. A partial fraction decomposition leads to a polynomial plus a term involving $1/(1 + x^2)$, which integrates immediately to the required result. An alternative is to use the substitution $x = \tan \theta$, leading to a polynomial in powers of $\tan \theta$. We then apply the recurrence relation for taking the integrals of powers of $\tan \theta$. Of course, the simplest approach today is to simply verify (1) using a symbolic manipulation package such as Maple or Mathematica. 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.