Conceptualizing definite integrals as ``signed area'' works great as long as one can actually visualize the ``area.'' In some cases, a better metaphor for integrals comes from the idea of {{CMD_87}}average value{{CMD_88}}. Looking back to your days as an even younger mathematician, you may recall the idea of an {{CMD_89}}average{{CMD_90}}: {{MATH_91}} If we want to know {{MATH_92}}, the average value of a function on the interval {{MATH_93}}, a naive approach might be to introduce {{MATH_94}} equally spaced grid points on the interval {{MATH_95}} {{MATH_96}} and choose a sample point {{MATH_97}} in each interval {{MATH_98}}, {{MATH_99}}.