Properties of Definite Integrals

By the end of this lesson, you will be able to apply all properties of definite integrals to simplify computations, use symmetry to evaluate integrals, and compute definite integrals using geometric area.

The Core Properties

For integrable functions $f$ and $g$ and constants $c$, $a$, $b$, $k$:

Zero integral: $\displaystyle\int_a^a f(x), dx = 0$ (no width → no area)

Reversal of limits: $\displaystyle\int_a^b f(x), dx = -\int_b^a f(x), dx$

Constant multiple: $\displaystyle\int_a^b kf(x), dx = k\int_a^b f(x), dx$

Sum/Difference: $\displaystyle\int_a^b [f(x) \pm g(x)], dx = \int_a^b f(x), dx \pm \int_a^b g(x), dx$

Additive over intervals: $\displaystyle\int_a^c f(x), dx = \int_a^b f(x), dx + \int_b^c f(x), dx$ (for any b)

Comparison: If $f(x) \leq g(x)$ on $[a,b]$, then $\displaystyle\int_a^b f(x), dx \leq \int_a^b g(x), dx$

Symmetry Properties

Even function ($f(-x) = f(x)$):

$$\int_{-a}^a f(x), dx = 2\int_0^a f(x), dx$$

Odd function ($f(-x) = -f(x)$):

$$\int_{-a}^a f(x), dx = 0$$

Examples:

• $\displaystyle\int_{-3}^3 x^4, dx = 2\int_0^3 x^4, dx = 2 \cdot \frac{3^5}{5} = \frac{486}{5}$ (even)
• $\displaystyle\int_{-\pi}^{\pi} \sin x, dx = 0$ (odd)
• $\displaystyle\int_{-2}^2 (x^3 + \cos x), dx = 0 + 2\int_0^2 \cos x, dx = 2\sin(2)$ (split odd + even parts)

Using Geometric Interpretation

When the integrand is a simple geometric shape:

$\displaystyle\int_0^3 \sqrt{9-x^2}, dx$ = area of quarter-circle with radius 3 = $\frac{1}{4}\pi(3)^2 = \frac{9\pi}{4}$

$\displaystyle\int_{-2}^4 f(x), dx$ where $f$ creates a triangle above the axis from 0 to 4 (area 4) and triangle below from −2 to 0 (area 1):

$= -1 + 4 = 3$ (signed area)

Building Integrals from Known Values

Example: Given $\displaystyle\int_0^5 f(x), dx = 7$ and $\displaystyle\int_3^5 f(x), dx = 2$, find $\displaystyle\int_0^3 f(x), dx$.

By the additive property: $\displaystyle\int_0^3 + \int_3^5 = \int_0^5$

$\displaystyle\int_0^3 f(x), dx = 7 - 2 = 5$