  We apply a result of David and Jon Borwein to evaluate a sequence of
highly-oscillatory integrals whose integrands are the products of a rapidly
growing number of sinc functions. The value of each integral is given in the
form $\pi(1-t)/2$, where the numbers $t$ quickly become very tiny. Using the
Euler-Maclaurin summation formula, we calculate these numbers to high
precision. For example, the integrand of the tenth integral in the sequence is
the product of 68100152 sinc functions. The corresponding $t$ is approximately
$9.6492736004286844634795531209398105309232 \cdot 10^{-554381308}$.
