The Integral

You know how fast you drove at every instant — your speedometer's reading. How far did you travel? You'd need to add up 'speed × tiny time' over every sliver of the trip — infinitely many of them. That accumulation is the integral. It's the second pillar of calculus, the mirror image of the derivative: where the derivative breaks a function into its instantaneous rate, the integral stitches infinitesimal pieces back into a total. Area under a curve, total distance from speed, volume of a solid, total charge from current — all integrals, all the same idea: add up the slices.