4-3: Riemann Sums & Definite Integrals Objectives: Understand the connection between a Riemann Sum and a definite integral Learn properties of definite integrals ©2003 Roy L. Gover (

Important Idea In the previous section, the width of all rectangles (  x ) was the same. This is not required; the rectangles can be different widths (  x i ).

Definition i th Rectangle cici f(c i ) A Riemann Sum Rectangle:

Definition The norm of the partition is the largest subinterval on x and is denoted by. If every subinterval is of equal width, the the partition is regular and is denoted by:

Important Idea For a given interval on the x axis, for example, from a to b,  0 implies n  As the width of the largest rectangle approaches 0, the number of rectangles approaches infinity a b

Important Idea As the width of the largest rectangle approaches 0, the number of rectangles approaches infinity a b

Important Idea As the width of the largest rectangle approaches 0, the number of rectangles approaches infinity a b

Important Idea As the width of the largest rectangle approaches 0, the number of rectangles approaches infinity a b

Analysis Compare and contrast:

Important Idea Letting the width of the widest rectangle is the same as letting the number of rectangles. As the width of each rectangle gets small, it makes no difference if you select l i, m i, or r i.

Definition If f is defined on the closed interval [ a,b ] and the limit exists, then f is integrable on [ a, b ] and the limit is denoted by:

Definition is called the definite integral from a to b where a is the lower limit of integration and b is the upper limit of integration

Example Evaluate the definite integral using a Riemann Sum:

Isn’t this fun!

Analysis How can area be negative? =-4.5 Can you write a definite integral that will calculate area of the red region?

Try This Evaluate the definite integral and find the area of the region using a Riemann Sum:, area is sq. units.

Definition If f is continuous and nonnegative on [ a,b ], then the area of the region bounded by the graph of f, the x axis and the vertical lines x=a & x=b is: a b

Important Idea The limit of a Riemann Sum is a definite integral which may be used to define the area of a region under a curve. Finding area is only one of many applications involving the limit of a sum.

Example Compare the definite integral and antiderivative:

Important Idea The definite integral is a number The indefinite integral is a family of functions

Try This Evaluate the definite integral using a Riemann Sum: 0

Definition Properties of Definite Integrals:

Definition Properties of Definite Integrals:

Definition Properties of Definite Integrals:

Definition Properties of Definite Integrals: Where k is a constant

Definition Properties of Definite Integrals:

Definition Properties of Definite Integrals: If f is nonnegative on [ a, b ], then

Definition Properties of Definite Integrals: If f(x)  g(x) for every x in [ a,b ], then

Try This Given and, find: 8

Try This Given and, find: -2

Try This Given and, find: 10

Lesson Close What is another name for the limit of a Riemann Sum?