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:
 * $\ds \int_a^b \map f x \rd x$

In the expression $\ds \int_a^b \map f x \rd x$, 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 $\ds \int_a^b \map f x \rd x$ 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 see
From the Fundamental Theorem of Calculus, we have that:


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

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

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