Materials play an important role in society. Historically, and even at present, materials have been discovered by trial and error, and many of the most useful materials have been discovered by chance. The high-throughput approach aims to remove (as much as possible) chance and guesswork at the experimental level by filtering out materials candidates that are not predicted to exist. Many successes have been recorded. In the high-throughput approach to materials discovery, machined-learned models of materials are created from databases of theoretical materials. These databases are the result of millions of density-functional-theory (DFT) simulations. The size and accuracy of the data in the databases (and, consequently, the predictions of machined-learned models) are most affected by the band energy calculation; most of the computation of a DFT simulation is computing the band energy in the self-consistency cycle, and most of the error in the simulation comes from band energy error. The band energy is obtained from a two-part multidimensional numerical integral over the Brillouin or irreducible Brillouin zone. A quadratic approximation and integration algorithm for computing the band energy in 2D and 3D is described. Analytic and semi-analytic integration of quadratic polynomials over simplices improves the accuracy and efficiency of the calculation. A method is proposed for estimating the error bounds of the quadratic approximation that does not require additional eigenvalues. Error propagation of approximation errors leads to an adaptive refinement approach that is driven by band energy error. Because adaptive meshes have little symmetry, integration is performed within the irreducible Brillouin zone, and a general algorithm for computing the irreducible Brillouin zone is described. The efficiency of quadratic integration is tested on realistic empirical pseudopotentials. When compared to current integration methods, uniform quadratic integration over the irreducible Brillouin zone sometimes results in fewer k-points for a given accuracy. Adaptive refinement fails to improve integration performance because band energy error bounds are inaccurate, especially at accidental crossings at the Fermi level.