Sec 5.2: The Definite Integral New Symbol Definition: the definite integral of ƒ over [a, b] Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Solution:

The procedure of calculating an integral is called integration. Sec 5.2: The Definite Integral Note 1: integrand limits of integration upper limit b lower limit a Integral sign The dx simply indicates that the independent variable is x. The procedure of calculating an integral is called integration.

the definite integral of f from a to b Sec 5.2: The Definite Integral Area under the curve Limit of the Riemann sum If you are asked to find one of them choose the easiest one. the definite integral of f from a to b three sides of the same coin

Example: Sec 5.2: The Definite Integral Evaluate the following integrals by interpreting each in terms of areas.

Sec 5.2: The Definite Integral the definite integral can be interpreted as the area under the curve definite integral has negative value A definite integral can be interpreted as a net area, that is,a difference of areas:

Example: Sec 5.2: The Definite Integral Evaluate the following integrals by interpreting each in terms of areas.

Example: Example: Sec 5.2: The Definite Integral Evaluate the following integrals by interpreting each in terms of areas. Example: Evaluate the following integrals by interpreting each in terms of areas.

Sec 5.2: The Definite Integral Term-121

the definite integral of f from a to b Sec 5.2: The Definite Integral Express the limit as a definite integral on the given interval. the definite integral of f from a to b

Sec 5.2: The Definite Integral Term-103

the definite integral of f from a to b Sec 5.2: 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.

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral Term-121

Sec 5.2: The Definite Integral In fact, instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number in the ith subinterval We call the numbers the sample points Definition: Definition: Area = Area =

Example: Sec 5.2: The Definite Integral Definition: Definition: Find the Riemann sum for ƒ(x) = x + 2 over [ 0, 5 ] divided into

the definite integral of f from a to b Sec 5.2: 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.

Sec 5.2: The Definite Integral Property (1) Example:

Sec 5.2: The Definite Integral Property (2)

Sec 5.2: The Definite Integral Property (3)

Sec 5.2: The Definite Integral Property (4) Property (5)

Sec 5.2: The Definite Integral Properties of the Integral

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral Definition: Example: provided that this limit exists Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Solution: Definition: If the limit does exist, we say that the function f is integrable the limit exist, is integrable

Theorem: Sec 5.2: The Definite Integral If f (x) is continuous on [a, b] f (x) is integrable Example: is not integrable in [0, 1] Remark f(x) has only finite number of removable discontinuities Remark f(x) has only finite number of jump discontinuities

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral Term-091

Sec 5.2: The Definite Integral Property

Sec 5.2: The Definite Integral Property (6)

Sec 5.2: The Definite Integral Property (7)

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral

Sec 5.2: The Definite Integral Term-082