# Definition:Definite Integral/Limits of Integration

## Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed interval of the set $\R$ of real numbers.

Let $f: \R \to \R$ be a real function.

Let $f \left({x}\right)$ be bounded on $\left[{a \,.\,.\, b}\right]$.

Let the definite integral of $f$ with respect to $x$ from $a$ to $b$ be:

- $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$

In the expression $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$, the values $a$ and $b$ are called the **limits of integration**.

If there is no danger of confusing the concept with limit of a function or of a sequence, just **limits**.

Thus $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$ can be voiced:

**The integral of (the function) $f$ of $x$ with respect to $x$ (evaluated) between the limits (of integration) $a$ and $b$.**

More compactly (and usually), it is voiced:

**The integral of $f$ of $x$ with respect to $x$ between $a$ and $b$**

or:

**The integral of $f$ of $x$ dee $x$ from $a$ to $b$**

From the Fundamental Theorem of Calculus, we have that:

- $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x = F \left({b}\right) - F \left({a}\right)$

where $F$ is a primitive of $f$, that is:

- $f \left({x}\right) = \dfrac {\mathrm d} {\mathrm d x} F \left({x}\right)$

Then $F \left({b}\right) - F \left({a}\right)$ is usually written:

- $\Big[{ F \left({x}\right) }\Big]_a^b := F \left({b}\right) - F \left({a}\right)$

or:

- $\Big.{ F \left({x}\right) }\Big|_a^b := F \left({b}\right) - F \left({a}\right)$

## Also known as

The interval defined by the **limits of integration** can be referred to as the **range of integration**.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore