Stacking Infinitesimals

Calculus 2 · Axiom Academy

The area under a curve isn't given by any formula you memorized — so we build it out of infinitely many infinitely thin slices, stacked into one exact total. That total is the integral. How do you measure a shape with a curved edge? Rectangles and triangles have area formulas. The region under a curve has no such formula — its top edge bends, so there is nothing to plug in. The whole idea of integration is a daring move around that wall: slice the region into strips so thin they are almost nothing, then add up infinitely many of them. You don't have to take it on faith — watch the area get built one sliver at a time. Watch the sweep line cross the curve from left to right. Everything it has passed fills in as area underneath, and the running total ∫ area climbs as more slivers are stacked on. When the sweep reaches the far edge, that accumulated area is the integral. The integral is just accumulated area — the running total the sweep line leaves behind as it stacks slice onto slice. Why thinner slices give a better answer Here is the same region cut into real, finite-width slices — a stack of rectangles whose tops just barely miss the curve. Drag the handle to chop it into more and more slices. Watch the jagged stack tighten onto the true curve, and watch the gap between the rectangle-sum and the exact area shrink toward zero as the slice width x does.

This is the written version of the interactive lesson above. See the full Calculus 2 course.