Area Under Curves

Real Analysis · Axiom Academy

A curved region has no neat area formula. So we cover it with rectangles, make them thinner, and let the total close in on the exact area — the integral. The area under a curve, made exact You can read the area of a rectangle or a triangle straight off a formula. But the shaded region under a curve , from x = 0.5 to x = 3.5 , has no formula waiting for it. The whole trick of integration is to build that area out of pieces simple enough to add — and then make the pieces vanish. Watch the sweep line cross the curve. 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 definite integral . The integral is just accumulated area — the running total the sweep line leaves behind, settling on 11.325. Cover it with rectangles — then shrink them Slide the handle to chop into more and more left-endpoint rectangles , each as tall as the curve at its left edge. Their total area is a Riemann sum. Watch the chunky staircase tighten onto the true curve, and the gap between the rectangle-sum and the exact area 11.325 shrink toward zero as the step . As the rectangle-sum becomes the integral — that's the limit at the heart of it.

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