Area Under the Curve

Calculus 1 · Axiom Academy

The big idea of integration: measure the area beneath a curve by slicing it into more and more pieces, until the answer becomes exact. How do you measure the area of a curved region? A rectangle, a triangle, a circle — geometry hands you a formula. But the region trapped between a curving graph and the axis has no formula waiting for it. The whole idea of integration is a way to get that exact area anyway, and it starts with a move you can watch happen. Watch the sweep line cross the curve from left to right: everything it passes fills in as area underneath, and the running total climbs toward the exact area beneath the whole curve. That accumulated area — the total the sweep line leaves behind — is what we call the integral. The integral is just accumulated area — the running total the sweep line leaves behind as it crosses the curve. Where that area comes from: rectangles that shrink Here is the trick that makes it work. Cover the region with thin rectangles, add up their areas, and you get an estimate. Slide the handle to chop it into more and more rectangles: the chunky staircase tightens onto the true curve, and the gap between your estimate and the real area vanishes as each piece gets narrower. As Δx → 0, the discrete sum becomes the continuous area — and that limit is the integral. The flip side: how fast the area is growing

This is the written version of the interactive lesson above. See the full Calculus 1 course.