Definition:Definite Integral/Limits of Integration

Definition
Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.

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

Let $\map f x$ be bounded on $\closedint a b$.

Let the definite integral of $f$ $x$ from $a$ to $b$ be:
 * $\displaystyle \int_a^b \map f x \rd x$

In the expression $\displaystyle \int_a^b \map f x \rd x$, 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 $\displaystyle \int_a^b \map f x \rd x$ can be voiced:
 * The integral of (the function) $f$ of $x$ $x$ (evaluated) between the limits (of integration) $a$ and $b$.

More compactly (and usually), it is voiced:
 * The integral of $f$ of $x$ $x$ between $a$ and $b$

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

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

Some sources refer to it as the interval of integration.

Also see
From the Fundamental Theorem of Calculus, we have that:


 * $\displaystyle \int_a^b \map f x \rd x = \map F b - \map F a$

where $F$ is a primitive of $f$, that is:
 * $\map f x = \dfrac \d {\d x} \map F x$

Then $\map F b - \map F a$ is usually written:
 * $\bigintlimits {\map F x} {x \mathop = a} {x \mathop = b} := \map F b - \map F a$

or, when there is no chance of ambiguity as to the independent variable:


 * $\bigintlimits {\map F x} a b := \map F b - \map F a$

Some sources use:
 * $\Big.{\map F x}\Big|_a^b := \map F b - \map F a$

but this is not recommended, as it is not so clear exactly where the expression being evaluated actually starts.