the.com/integral
the math of slicing infinity into pieces thin enough to actually add up
means Essential to the whole, so deeply built-in that the thing falls apart without it — or, in mathematics, the operation that sums up infinitely many infinitely small slices to find a total.
from From Latin 'integer,' meaning whole, untouched, complete — the same root that gives us 'integer' (a whole number) and 'integrity' (a whole, unbroken character). It traces back to 'in-' (not) plus a relative of 'tangere' (to touch), so an integral is literally something 'untouched,' intact, all of a piece. The mathematical sense was settled in the late 1600s as calculus took shape: a quantity built by gathering all its parts back into a whole.
symbol originThat stretched S means summa, Latin for sum
Newton vs LeibnizBoth invented it; they feuded for decades
area trickAdds infinitely many rectangles of zero width
reverse gearIt perfectly undoes the derivative
everyday useComputes distances, volumes, probabilities, electrical charge