As computer hardware and software continue to improve, we are able to perform increasingly complex calculations. By the end of this decade the computational science community is aiming to achieve $10^{18}$ floating point operations per second. Extreme scale computing will be used for accurately simulating and understanding the world's climate and extracting meaningful information from humongous amounts of data, among other applications. \emph{Reliable} extreme scale computing assumes the reliability of the component algorithms. This talk focuses on some simpler numerical algorithms that we take for granted, such as those behind the \texttt{sin} buttons on our calculators and those that compute definite integrals. Although these algorithms have been developed over decades, it is not commonly understood under what conditions they may fail. This talk provides some examples of dramatic failure. We also demonstrate how to construct a reliable, adaptive numerical integration algorithm based on the trapezoidal rule. A key idea is to focus on a cone of integrands rather than a ball. We briefly explain how the idea of a cone can be applied to develop other reliable numerical algorithms. The Guaranteed Automatic Integration Library (GAIL) is a MATLAB software package that we have developed to make our new, reliable algorithms widely available.