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.
Appears in: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration