4-3: riemann sums & definite integrals objectives: understand the connection between a riemann...
TRANSCRIPT
4-3: Riemann Sums & Definite IntegralsObjectives:•Understand the connection between a Riemann Sum and a definite integral•Learn properties of definite integrals©2003 Roy L. Gover
(www.mrgover.com)
Important IdeaThe 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.
Important Idea
•The definite integral is a number•The indefinite integral is a family of functions
DefinitionProperties of Definite Integrals:
( ) 0a
a
f x dx
DefinitionProperties of Definite Integrals:
( ) ( )a b
b a
f x dx f x dx
DefinitionAdditive Interval Property: If f is
integrable on the three closed intervals determined by a, b, and c, then
( )b
a
f x dx
( ) ( )c b
a c
f x dx f x dx
DefinitionProperties of Definite Integrals:
( ) ( )b b
a a
kf x dx k f x dx Where k is a constant
DefinitionProperties of Definite Integrals:
( ) ( )b
a
f x g x dx ( ) ( )
b b
a a
f x dx g x dx
DefinitionProperties of Definite Integrals:
If f is nonnegative on [a,b], then
( ) 0b
a
f x
DefinitionProperties of Definite Integrals:If f(x)g(x) for every x in [a,b], then
( ) ( )b b
a a
f x g x
Try This
Given and ,find:
3
0
( ) 2f x 8
3
( ) 6f x dx 8
0
( )f x dx8
Try This
Given and ,find:
3
0
( ) 2f x 8
3
( ) 6f x dx 0
3
( )f x dx-2
Try This
Given and ,find:
3
0
( ) 2f x 3
0
( ) 6g x dx 3
0
[2 ( ) ( )]f x g x dx10
Lesson Close
What is another name for the limit of a Riemann Sum?
Assignment
Page 278
Problems 13 – 23, 25, 27, 33 – 43 odd, 47, 48, 49