Riemann Sum. Formula Step 1 Step 2 Step 3 Riemann Sum.

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Presentation transcript:

Riemann Sum

Formula

Step 1 Step 2 Step 3 Riemann Sum

We start by subdividing the interval [a,b] into n subintervals The width of the interval [a,b] is b-a the width of each subinterval is The subintervals are Riemann Sum

Term-103

The Definite Integral Sec 5.3

Remark: Definition: the definite integral of ƒ over [a, b] The Definite Integral Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Definition: the definite integral of ƒ over [a, b]

Partition is called a partition of [a, b]. Example Is a partition of [0, 10]. Is a partition of [0, 9]. Is a partition of [0, 10]. Note that the length of subintervals are not the same

Example Is a partition of [0, 10]. the largest of all the subinterval widths subinterval widths Partition Def:Norm of the partition

Riemann sum for ƒ on the interval [a, b]. Riemann Sum Example: Find the Riemann sum for ƒ(x) = x + 2 over [ 0, 5 ]

Definition: The Definite Integral the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b]

Notation: the definite integral of ƒ over [a, b] Remark: The Definite Integral Remark:

The Definite Integral

Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral

Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral

Area under the curve the definite integral of f from a to b If you are asked to find one of them choose the easiest one. The Definite Integral

Riemann sum for ƒ on the interval [a, b].

Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral

THE DEFINITE INTEGRAL Term-103

Property (1) THE DEFINITE INTEGRAL Example:

THE DEFINITE INTEGRAL Property (2)

THE DEFINITE INTEGRAL Property (3)

THE DEFINITE INTEGRAL Term-091

THE DEFINITE INTEGRAL

DEFINITION Example: Find the average value of the function over the interval [-2,2] Average Value

Term-082