Riemann Sums and Definite Integrals for AP Calculus

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 x0 =a, xn =b anda < x₁ < x2 < x₃ …< x_n–1 < b. The points a, x₁, x₂, x₃, … x_{n +1}, b form a partitionof f denoted as Δ on [a, b]. Let Δxi be the length of the ith interval [x_i–1, xi ] and c i beany point in the ith 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 Δx_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

1. If f is defined on [a, b], and the limit exists, then f is integrable on [a, b].
2. If f is continuous on [a, b], then f is integrable on [a, b].

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

3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. provided g (x ) ≤ f (x ) ≤ h(x) on [a, b].
11. 
12. f (x )dx; provided f (x ) is integrable on an interval containing a, b, c.

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