2.4.3—How the area under a curve

The signed area between y=f(x) and the x-axis from a to b is ∫_a^b f(x)dx. Geometric area requires splitting where f changes sign or integrating top minus bottom between curves.

Find intersections first and use the interval that traces the intended region once. In parametric or polar forms, change the area formula rather than forcing Cartesian dx.

If f is below the x-axis on [a,b], the geometric area is −∫_a^b f(x)dx. For two curves, integrate (upper−lower) only after checking which is larger throughout.

A definite integral can be zero while geometric area is positive because positive and negative signed regions cancel.