Transcript Riemann Sums and Definite Integrals

Definite Integrals
Section 4.3
1
The Definite Integral
The definite integral as the area of a region:
If f is continuous and non-negative on the closed interval [a, b],
then the area of the region bounded by the graph of f, the
x-axis, and the vertical lines x = a and x = b is given by
b
Area =
 f ( x)dx
a
f
This is called the definite integral.
A

b
a
f ( x)dx  A
a
b
2
The Definite Integral
b
n
a
i 1
f (ci )x

 f ( x)dx  nlim

where ci is any point in the ith interval andx 
ba
n
********************************************
At this point, we evaluate a definite integral using area
formulas of common geometric regions, if possible.
(In the next section, we will calculate the definite integral using other methods)
3
Using Common Geometric Figures
Example:
5
 3dx
1
4
Using Common Geometric Figures
Example:
1
 (3  2 x)dx
2
5
Using Common Geometric Figures
Example:
5
  5  x  dx
5
6
Using Common Geometric Figures
Example:
4

16  x 2 dx
4
7
Properties of the Definite
Integral
f
a

a

a
a
b

b
a
b
c
f ( x)dx  0 by definition
b
f ( x)dx    f ( x)dx
a
f ( x)dx 

c
a
by definition
b
f ( x)dx   f ( x)dx
c
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Definite Integrals
f
A1
A3
a
b
A2

b
a
f ( x)dx  A1  A3  A2
= area above – area below
9
Properties…

b
a
b
kf ( x)dx  k  f ( x)dx
a
  f ( x)  g ( x)dx 
b
b
a
a
b
f ( x)dx   g ( x)dx
a
10
Example
If

5
2
f ( x)dx  3
then find
and

9
2
f ( x)dx  10,
5
 4 f ( x)dx.
9
11
Example
Set up a definite integral that yields the area of the region.
f ( x)  5  x2
12
Example
Set up a definite integral that yields the area of the region.
f ( x)  cos x


2

2
13
Homework
Section 4.3 page 278 # 13 – 19 odd, 23 – 43 odd
14