AP Calculus AB and BC glossary
Area under a curve
The area under a curve is the region trapped between the graph and the x-axis over an interval. Where the function is non-negative it equals the definite integral exactly; where the function dips below the axis you must integrate the absolute value instead.
The idea comes from Riemann sums: chop the region into rectangles, add their areas, and take the limit as the rectangles get thin. The integral is the exact value that process converges to.
The mistake
Applying the formula when the curve crosses the axis. Check the sign of the function on the interval before equating area with the integral.
Physics leans on this more than calculus does: the area under a velocity-time graph is displacement, under a force-position graph it is work, and under a force-time graph it is impulse. PhysicsLearn tabulates which graph gives which in area under a curve.
The statistics course is built on this idea under another name. For a continuous random variable, the area under the density curve across an interval IS the probability of landing in it, which is also why a single point has probability 0: an interval of zero width encloses no area.
Appears in: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration