Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

In this study guide:

- Sigma Notation or Summation Notation
- Definition of a Riemann Sum
- Definition of a Definite Integral
- Properties of Definite Integrals

### Sigma Notation or Summation Notation

where *i* is the index of summation, *l* is the lower limit and *n* is the upper limit of summation.

(Note: The lower limit may be any non-negative integer ≤ *n*.)

### Examples

### Summation Formulas

If *n* is a positive integer, then:

### Example

(Note: This question has not appeared in an AP Calculus AB Exam in recent years).

### Definition of a Riemann Sum

Let *f* be defined on [*a*, *b*] and *xi* 's be points on [*a*, *b*] such that *x*0 =*a*, *xn* =*b* and*a* < *x*_{1} < *x**2* < *x*_{3} …< *x*_{n–1} < *b*. The points *a*, *x*_{1}, *x*_{2}, *x*_{3}, … *x*_{n +1}, *b* form a partitionof *f* denoted as Δ on [*a*, *b*]. Let Δ*xi* be the length of the ith interval [*x*_{i–1}, *xi* ] and *c* i beany point in the *i*th interval. Then the Riemann sum of *f* for the partition is

### Example 1

Let *f* be a continuous function defined on [0, 12] as shown below.

Find the Riemann sum for *f* (*x* ) over [0, 12] with 3 subdivisions of equal length and the midpoints of the intervals as *c*_{i} '_{s}.

Length of an interval (See Figure 11.1-1.)

Riemann sum=

=7(4)+39(4)+103(4)=596

The Riemann sum is 596.

### Example 2

Find the Riemann sum for *f* (*x* )=*x* 3 +1 over the interval [0, 4] using 4 subdivisions of equal length and the midpoints of the intervals as *c*_{i} '*s*. (See Figure 11.1-2.)

Length of an interval

### Definition of a Definite Integral

Let *f* be defined on [*a, b*] with the Riemann sum for *f* over [*a, b*] written as

If max Δ*x*_{i} is the length of the largest subinterval in the partition and the exists, then the limit is denoted by:

*f*(*x* )*dx* is the definite integral of *f* from *a* to *b*.

#### Example 1

Use a midpoint Riemann sum with three subdivisions of equal length to find the approximate value of

midpoints are *x* =1, 3, and 5.

#### Example 2

Using the limit of the Riemann sum, find

Using *n* subintervals of equal lengths, the length of an interval

(Note: This question has not appeared in an AP Calculus AB Exam in recent years.)

### Properties of Definite Integrals

- If
*f*is defined on [*a, b*], and the limit exists, then*f*is integrable on [*a, b*]. - If
*f*is continuous on [*a, b*], then*f*is integrable on [*a, b*]. - provided
*g*(*x*) ≤*f*(*x*) ≤*h*(*x*) on [*a, b*]. -
*f*(*x*)*dx*; provided*f*(*x*) is integrable on an interval containing*a, b, c*.

If *f* (*x* ), *g* (*x* ), and *h*(*x* ) are integrable on [*a, b*], then

#### Examples

The remaining properties are best illustrated in terms of the area under the curve of the function as discussed in the next section.

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

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