Riemann Sum Riemann Sum Formula Riemann Sum Step

Riemann Sum

Riemann Sum Formula

Riemann Sum 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

Sec 5. 3 The Definite Integral

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

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

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

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

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

The Definite Integral 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.

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

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

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

THE DEFINITE INTEGRAL Term-103

THE DEFINITE INTEGRAL Property (1) Example:

THE DEFINITE INTEGRAL Property (2)

THE DEFINITE INTEGRAL Property (3)

THE DEFINITE INTEGRAL Term-091

THE DEFINITE INTEGRAL

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

Term-082