Definition of a Definite Integral

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Let
*f* be defined on the closed interval and divide this interval into *n* subintervals of equal width .
Let denote the endpoints of these subintervals and
choose **sample points** in these subintervals so that lies in the *i*th subinterval .
Then the **definite integral of f from a to b** (if it exists)
is

**Note:** The definite integral is a number; it does
not depend on *x*.

**Note:** If *f *is continuous it can be shown that the above limit exists no
matter how the sample points are chosen. It is convenient to take the sample
points to be the right endpoints .
Then the definition of an integral becomes

**Note:** The sum

is
called a **Riemann sum**. If *f* happens to be positive, this sum can
be interpreted as a sum of areas of approximating rectangles. In fact, in this
case (with *f *continuous on ), the definite integral can be interpreted as the area under the curve
from *a*
to *b*.

If *f* takes on positive and negative values on ,
then the definite integral can be interpreted as a **net area**, the
difference between the area above the *x*-axis
and the area below :

**Note:** The interval [*a*, *b*] need not be divided
into subintervals of equal length in order to define the definite integral;
using subintervals of equal length just makes things easier.

As **examples**,
evaluate the
following definite integrals.

For the second example above, first evaluate a Riemann sum taking the sample points to be right endpoints and , , and .

Two Special Definite Integrals

If *f* is defined at ,
then .

If *f* is integrable on ,
then .

**Preservation of Inequality.** If *f* and *g* are integrable on
and for every *x*
in ,
then

**One Final Important
Property:**

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