The Definite Integral as Area

Calculus 1 · Axiom Academy

LESSON The Definite Integral as Area How the definite integral measures the signed area between a curve and the x-axis — and why everything below the axis counts as negative. 1. The Integral Is Area, Accumulating For a continuous function f on [a, b] , the definite integral is the area between the curve y = f(x) and the x-axis from x = a to x = b . Picture a vertical line sweeping left to right: everything it has already passed fills in underneath, and that running total is the integral so far. the limits a and b say where to start and stop measuring 2. Above the Axis Adds, Below Subtracts Here is the twist. When f(x) > 0 the strips have positive height and the integral grows. When f(x) < 0 the strips have negative height, so they pull the total back down. Watch the running area below dip negative as the sweep crosses under the axis, then climb back as it rises above. On [2, 4] the line f(x) = x - 2 sits above the axis. That triangle has area 2 , and . On [0, 2] the same line is below the axis. Its triangle also has area 2 , but . 3. Net Signed Area: When the Pieces Cancel When a curve crosses the axis, the integral over the whole interval reports the net result: the area above minus the area below. For our line on [0, 4] , the +2 above exactly cancels the -2 below, so the total is 0 — even though there is real area on both sides. What the definite integral actually returns. The actual amount of region, found by integrating |f(x)| instead.

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