Definition:Definite Integral

Definition

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

Let $f: \left[{a \, . \, . \, b}\right] \to \R$ be a function.

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

Suppose that:

$\displaystyle \sup_P L \left({P}\right) = \inf_P U \left({P}\right)$

where the supremum and infimum are taken over all subdivisions $P$ of $\left[{a \, . \, . \, b}\right]$, and $L \left({P}\right)$ and $U \left({P}\right)$ denote the lower sum and upper sum of $f \left({x}\right)$ on $\left[{a \, . \, . \, b}\right]$ belonging to the subdivision $P$, respectively.

Then the definite (Riemann) integral of $f \left({x}\right)$ over $\left[{a \, . \, . \, b}\right]$ is defined as:

$\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \sup_P L \left({P}\right) = \inf_P U \left({P}\right)$

$f \left({x}\right)$ is formally defined as (properly) integrable over $\left[{a \, . \, . \, b}\right]$ in the sense of Riemann or Riemann integrable.

More usually (and informally), we say:

$f \left({x}\right)$ is integrable over $\left[{a \, . \, . \, b}\right]$.

If $a > b$ then we define:

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

Limits of Integration

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:

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

or:

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

Geometric Interpretation

The expression $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ can be (and frequently is) interpreted as the area under the graph. This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the "height" of the function being integrated in that interval and the formula for the area of a rectangle.

A depiction of the lower and upper sums illustrates this:

It can intuitively be seen that as the number of points in the subdivision increases, the more "accurate" the lower and upper sums become.

Also note that if the graph is below the $x$-axis, the signed area under the graph becomes negative.

Integrand

In the expression $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$, the function $f \left({x}\right)$ is called the integrand.

This term comes from the cod-Latin for that which is to be integrated.

Historical Note

Consider the Riemann sum:

$\displaystyle \sum_{i=1}^n \ f\left({c_i}\right) \ \Delta x_i$

Historically, the definite integral was an extension of this type of sum such that:

• The finite distance $\Delta x$ is instead the infinitely small distance $\mathrm dx$

• The finite sum $\Sigma$ is instead the sum of an infinite amount of infinitely small quantities: $\int$

Hence the similarity in notation:

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \sum_a^b \ f\left({x}\right) \ \Delta x\)

\(\to\)

\(\displaystyle \int_a^b f\left({x}\right) \ \mathrm dx\)

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \)

as $\Delta x \to \mathrm dx$

The notion of "infinitely small" does not exist in the modern formulation of real numbers. Nevertheless, this idea is sometimes used as an informal interpretation of the definite integral.

Also see

• Riemann sum
• Signed area

Note that a continuous function is always Riemann integrable.

There are more general definitions of integration; see Lebesgue Integral is Extension of Riemann Integral.

Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 13.2$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 13.16$