# Properties of Definite Integrals: Rules, Symmetry and Examples

> The properties of definite integrals: reversing limits, zero width, additivity over intervals, odd and even symmetry and comparison bounds, with examples.

Source: https://practicecalculus.com/rules/definite-integral-properties
Updated: 2026-11-03

$$ \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx $$

## At a glance

- **Zero width:** $\int_a^a f(x)\,dx = 0$
- **Reversed limits:** $\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$
- **Constant:** $\int_a^b k\,dx = k(b - a)$
- **Odd function:** $\int_{-a}^{a} f(x)\,dx = 0$
- **Even function:** $\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$
- **Comparison:** If $f(x) \le g(x)$ on $[a, b]$, then $\int_a^b f\,dx \le \int_a^b g\,dx$

## Key takeaways

- Swapping the limits of integration changes the sign of the integral.
- You can split an integral at any point $b$: $\int_a^c = \int_a^b + \int_b^c$.
- Over a symmetric interval, an odd function integrates to $0$ and an even function to twice the right half.
- If $m \le f(x) \le M$ on $[a, b]$, then the integral lies between $m(b-a)$ and $M(b-a)$.

The **properties of definite integrals** are the rules for manipulating $\int_a^b f(x)\,dx$ without computing it from scratch: flipping limits, splitting intervals, using symmetry and estimating size. They all follow from viewing the integral as signed area, or from the [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus).

## The properties

Assume $f$ and $g$ are integrable on the intervals involved.

| Property | Statement |
| --- | --- |
| Zero width | $\int_a^a f(x)\,dx = 0$ |
| Reversed limits | $\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$ |
| Constant | $\int_a^b k\,dx = k(b-a)$ |
| Linearity | $\int_a^b [\alpha f + \beta g]\,dx = \alpha\int_a^b f\,dx + \beta\int_a^b g\,dx$ |
| Additivity | $\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$ |
| Comparison | $f \le g$ on $[a,b]$ $\Rightarrow$ $\int_a^b f\,dx \le \int_a^b g\,dx$ |
| Bounds | $m \le f \le M$ on $[a,b]$ $\Rightarrow$ $m(b-a) \le \int_a^b f\,dx \le M(b-a)$ |
| Odd symmetry | $f(-x) = -f(x)$ $\Rightarrow$ $\int_{-a}^{a} f\,dx = 0$ |
| Even symmetry | $f(-x) = f(x)$ $\Rightarrow$ $\int_{-a}^{a} f\,dx = 2\int_0^a f\,dx$ |

The comparison and bounds properties assume $a < b$. Additivity holds for **any** order of $a$, $b$ and $c$, as long as $f$ is integrable on the intervals, thanks to the reversed-limits rule.

## Why they hold

If $F$ is an antiderivative of $f$, then $\int_a^b f\,dx = F(b) - F(a)$. Each property is a line of algebra from there:

- zero width: $F(a) - F(a) = 0$;
- reversed limits: $F(a) - F(b) = -\left(F(b) - F(a)\right)$;
- additivity: $\left(F(b) - F(a)\right) + \left(F(c) - F(b)\right) = F(c) - F(a)$.

Linearity is covered on its own page: [integral linearity rules](/rules/integral-linearity-rules).

## Symmetry: odd and even functions

```graph
title: 'Signed area of the odd function y = x³ − 2x on [−1.5, 1.5]'
caption: 'The area on the left is the mirror image of the area on the right with the opposite sign, so $\int_{-1.5}^{1.5} (x^3 - 2x)\,dx = 0$.'
x: [-2, 2]
y: [-2, 2]
functions:
  - expr: x^3 - 2x
    label: y = x³ − 2x
shade: {expr: x^3 - 2x, from: -1.5, to: 1.5}
```

An odd function is symmetric about the origin, so every bit of positive area has a matching bit of negative area. An even function is symmetric about the $y$-axis, so the two halves are equal and you can double one of them. Checking symmetry first can turn a long calculation into one line.

## Worked examples

**Example 1: odd symmetry.** $x^5 - 4x^3 + \sin x$ is odd, so

$$\int_{-3}^{3} \left(x^5 - 4x^3 + \sin x\right)dx = 0$$

**Example 2: even symmetry.** $x^2$ is even, so

$$\int_{-1}^{1} x^2\,dx = 2\int_0^1 x^2\,dx = 2\cdot\frac{1}{3} = \frac{2}{3}$$

**Example 3: additivity and reversed limits.** Given $\int_0^5 f(x)\,dx = 10$ and $\int_0^2 f(x)\,dx = 3$:

$$\int_2^5 f(x)\,dx = 10 - 3 = 7, \qquad \int_5^2 f(x)\,dx = -7$$

**Example 4: an absolute value.** Split where the inside changes sign. On $[-1, 0]$, $\lvert x\rvert = -x$; on $[0, 2]$, $\lvert x\rvert = x$:

$$\int_{-1}^{2} \lvert x\rvert\,dx = \int_{-1}^{0} (-x)\,dx + \int_0^2 x\,dx = \frac{1}{2} + 2 = \frac{5}{2}$$

**Example 5: estimating with bounds.** On $[0, 1]$, $e^{-1} \le e^{-x^2} \le 1$, so

$$0.37 \approx e^{-1} \le \int_0^1 e^{-x^2}\,dx \le 1$$

This integral has no elementary antiderivative, yet the bounds pin it down. Its actual value is about $0.747$.

## Common mistakes

1. **Using symmetry on a non-symmetric interval.** $\int_{-1}^{2} x^3\,dx$ is not $0$. The interval must be $[-a, a]$.
2. **Calling a function odd without checking.** $x^3 + 1$ is neither odd nor even, so $\int_{-1}^{1} (x^3 + 1)\,dx = 2$, not $0$.
3. **Forgetting the sign flip** when reversing limits.
4. **Confusing signed area with total area.** $\int_{-1}^{1} x\,dx = 0$, but the total shaded area is $1$. For area, integrate $\lvert f(x)\rvert$, as in [area between curves](/blog/area-between-curves).

For how these properties connect to Riemann sums and evaluating integrals, see the guides to [Riemann sums](/blog/riemann-sums) and the [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus).

## Frequently asked questions

### What happens when you switch the limits of a definite integral?

The integral changes sign: $\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$. This follows directly from evaluating $F(a) - F(b)$ instead of $F(b) - F(a)$.

### What is the integral of an odd function from -a to a?

It is $0$, because the signed area on the left exactly cancels the signed area on the right. For example $\int_{-2}^{2} x^3\,dx = 0$.

### Can you split a definite integral into two?

Yes. For any $b$, $\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx$. This is how you integrate piecewise functions and absolute values.

### Is the integral from a to a always zero?

Yes. An interval of width zero has no area, so $\int_a^a f(x)\,dx = 0$ for any function $f$ defined at $a$.