SL 5.11—Definite integrals and areas

A definite integral accumulates signed change over an interval.

∫ₐᵇf(x)dx is net signed area; geometric area may require splitting where f changes sign.

Example

∫₋¹¹x dx=0 by cancellation, although the total geometric area is 1.

Find roots and compare the requested quantity—net area, total area or accumulated change.

A negative integral is possible; it does not mean an area has become physically negative.

Fundamental evaluation: if $F'=f$, then $\int_a^b f(x)\,dx=F(b)-F(a)$. For area between $y=f(x)$ and $y=g(x)$, first find intersections, then integrate top minus bottom on each interval: $A=\int_a^b|f(x)-g(x)|\,dx$, splitting wherever their order changes. This preserves geometric area even when the signed integral cancels.