Properties of Definite Integrals

By the end of this lesson, you will be able to apply all properties of the definite integral to evaluate and estimate integrals, use symmetry and comparison properties.

Properties of the Definite Integral

If $f$ and $g$ are integrable on $[a,b]$:

1. Zero interval: $\int_a^a f(x),dx = 0$

2. Reversal: $\int_a^b f,dx = -\int_b^a f,dx$

3. Sum: $\int_a^b[f \pm g],dx = \int_a^b f,dx \pm \int_a^b g,dx$

4. Constant: $\int_a^b kf,dx = k\int_a^b f,dx$

5. Splitting: $\int_a^b f,dx = \int_a^c f,dx + \int_c^b f,dx$ (for any $c$)

6. Comparison: if $f(x) \leq g(x)$ on $[a,b]$: $\int_a^b f,dx \leq \int_a^b g,dx$

7. Bounding: if $m \leq f(x) \leq M$ on $[a,b]$: $m(b-a) \leq \int_a^b f,dx \leq M(b-a)$

Symmetry Properties

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

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

Example 1 — Using Splitting Property

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

$\int_0^5 f,dx = \int_0^3 f,dx + \int_3^5 f,dx$

$7 = \int_0^3 f,dx + 2 \implies \int_0^3 f,dx = 5$

Example 2 — Symmetry

$$\int_{-3}^3 (x^4 - 3x^2 + 5),dx$$

The integrand is even (all even powers):

$= 2\int_0^3(x^4 - 3x^2 + 5),dx = 2\left[\frac{x^5}{5} - x^3 + 5x\right]_0^3$

$= 2\left[\frac{243}{5} - 27 + 15\right] = 2\left[\frac{243}{5} - 12\right] = 2\cdot\frac{183}{5} = \frac{366}{5}$

Versus: $\int_{-3}^3(x^3 - 7x),dx = 0$ (odd function).

Example 3 — Bounding an Integral

Bound $\int_0^1 e^{-x^2},dx$ without computing it.

On $[0,1]$: $e^{-1} \leq e^{-x^2} \leq e^0 = 1$ (since $x^2 \in [0,1]$ and $e^{-x}$ is decreasing)

$e^{-1}(1) \leq \int_0^1 e^{-x^2},dx \leq 1(1)$

$0.368 \leq \int_0^1 e^{-x^2},dx \leq 1$

(Actual: $\approx 0.747$)

Example 4 — Combining Properties

$$\int_2^5 [3f(x) - 2g(x)],dx = 3\int_2^5 f(x),dx - 2\int_2^5 g(x),dx$$