Properties Of Integrals

Properties of Integrals

The definite integral $\int_a^b f(x)\,dx$ represents signed area and obeys fundamental algebraic properties—much like sums and limits. These structural rules let you evaluate complex integral expressions using only given values, without computing a single antiderivative.

Linearity

The integral is linear: constants factor out and integrals of sums split apart: $\int_a^b c\,f(x)\,dx = c\int_a^b f(x)\,dx$ $\int_a^b \big(f(x) \pm g(x)\big)\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$

This mirrors linearity of sums, since the definite integral itself is defined as a limit of Riemann sums.

Bounds Rules

When the upper and lower bounds are equal, there is no interval: $\int_a^a f(x)\,dx = 0$

Reversing the bounds flips the sign: $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

Additivity Over Intervals

For points $a < b < c$, the integral over the full interval splits at any interior point: $\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx$

Geometrically, total area equals the sum of parts. This is especially powerful when a function changes formula or sign.

Comparison Property

If $f(x) \ge 0$ on $[a,b]$: $\int_a^b f(x)\,dx \ge 0$

More generally, if $f(x) \ge g(x)$ on $[a,b]$: $\int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx$

This lets you compare integrals without computing them exactly, since each Riemann sum term for $f$ is at least as large as the corresponding term for $g$.