Area Under the Curve

Calculus 1 · Axiom Academy

From a fistful of rectangles to a single exact number — the definite integral. How do you measure the area under a curve? A rectangle is easy: base times height. But the region trapped under a curving graph has no clean formula — its top is bending the whole way across. The whole idea of integration is to corner that area anyway, by accumulating it a sliver at a time. Watch the sweep line cross the parabola y = x^2 from x=0 to x=2 . Everything it passes fills in as area underneath, and the running total climbs to the exact area beneath the whole curve. That accumulated area is the integral. The integral is just accumulated area — the running total the sweep line leaves behind, settling on the exact value 8/3 2.667. Where that area comes from: rectangles Here's the trick made concrete. Chop [0,2] into N equal strips and cap each one with a rectangle as tall as the curve at the strip's right edge. Add up the rectangles and you approximate the area. Slide N up and watch the staircase tighten onto y=x^2 — the total marching toward the exact . As x → 0, the rectangles become the region itself — that limit is the exact area. You just did both halves of one idea: pile up rectangles, then shrink them without bound. In the limit, the approximation stops being an approximation and becomes the exact area. That limit has a name and a symbol.

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