6.6 Applying Properties of Definite Integrals

Definite integrals obey several important algebraic and geometric properties. For example, ${\int_a^a f(x)\,dx=0}$ because no interval means no accumulation, and reversing the limits changes the sign: ${\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx}$. Integrals are also additive over intervals, so ${\int_a^b f(x)\,dx+\int_b^c f(x)\,dx=\int_a^c f(x)\,dx}$. In addition, constants can be factored out and sums or differences can be integrated term by term. These properties make definite integrals easier to work with even before you know many antiderivative rules.

You should think of these properties as natural consequences of accumulation. If you accumulate from a to b and then from b to c, that should equal the total accumulation from a to c. If you go backward across an interval, the sign flips. If a function is split into simpler parts, the accumulated totals also split accordingly. These rules allow simplification, help with evaluation, and make it easier to reason about integrals geometrically and algebraically in later problems.