Adding Infinitely Small Pieces

Integration is the art of summing infinitely many tiny slices to find a total. Need the area under a curve? Slice it into ultra-thin rectangles, add their areas, and let the width shrink toward zero. The result is exact, not an approximation.

This idea solved problems that stumped mathematicians for millennia. Archimedes estimated the area of a circle using polygons with more and more sides — a 96-sided polygon got him within 0.04% of the true value. Integration formalizes that instinct: smaller pieces for better accuracy, then take the limit.

The leap from clever geometry to formal integration took roughly 1,900 years, bridged at last by Newton and Leibniz in the late 1600s.