IntegratIon, fInIte sum and defInIte
Integral
1
.
A
Figure 5.1.8
Figure 5.1.9
Upper and lower estimates of the area in Example 2
A
A
Approximations of the area for
n = 2, 4, 8 and 12 rectangles
Section 1 / Figure 1
(a ) n  2
( c) n  8
( b) n  4
(d) n  12
Approximating rectangles when sample points are not endpoints
5. 1 Sigma notation
1 2  3  4  5 
5
k
k 1
6
1  8  27  64  125  216   i
3
i 1
4
k
1
2
3
4 163
 k  1  2  3  4  5  60
k 1
3

k 1
f ( xk ) xk  f ( x1 ) x1  f ( x2 ) x2  f ( x3 ) x3
The graph of a typical function y = ƒ(x) over [a, b].
Partition [a, b] into n subintervals a < x1 < x2 <…xn <
5.2
b. Select a number in each subinterval ck. Form the
product f(ck)xk. Then take the sum of these products.
n
 f (ck )xk
k 1
The curve of with rectangles from finer partitions
of [a, b]. Finer partitions create more rectangles
with
n shorter bases.
 f (ck )xk
k 1
n
lim P 0  f (ck )xk
k 1
The definite integral of f(x) on [a, b]
n
lim P 0  f (ck )xk
k 1
b

f ( x)dx
a
If f(x) is non-negative, then the definite integral represents the
area of the region under the curve and above the x-axis
between the vertical lines x =a and x = b
Rules for definite integrals
b

a
f ( x)dx    f ( x )dx
a
a

a
b
b
f ( x)dx  0
b
 kf ( x)dx  k 
a
a
f ( x)dx
a
a
 ( f ( x)  g ( x))dx  
b
b
c
a
a
a
f ( x) dx   g ( x) dx
b
 ( f ( x))dx  
b
b
f ( x) dx   f ( x) dx
c
5.3 The Fundamental Theorem of Calculus
x
1. If g ( x)   f (t )dt ,
then g ( x)  f ( x)
a
Where f(x) is continuous on [a,b] and differentiable on (a,b)
x
d
f (t )dt  f ( x)

dx a
Find the derivative of the function:
x
g ( x)   (t 3  5t  sin t )dt
a
The Fundamental Theorem of Calculus
b
2.
 f ( x)dx  F (b)  F (a)
a
4
2
(
x
  3x)dx
2

3
2
sec
  d
0
where F ( x)  f ( x)
Indefinite Integrals
 f ( x)dx  F ( x)
means
F ( x)  f ( x)
Definite Integrals

b
a
f ( x)dx  F (b)  F (a)
where F ( x)  f ( x)
Try These
1)  (2sin   5cos   3sec 2  )d
3y  2
2) 
dy
y
3
3)  (1  x 2 )dx
1
Answers
1)  (2sin   5cos   3sec2  )d
2cos   5sin   3tan   C
1
1

3y  2
2) 
dy   (3 y 2  2 y 2 )dy
y
3
2
1
2
2y  4y  C
3
3
1
1
3)  (1  x 2 ) 2 dx   (1  2 x 2  x 4 )dx
2 3 1 5 3
[ x  x  x ]1  69.6  1.8667  67.733
3
5
Indefinite Integrals and Net Change
• The integral of a rate of change is the net change.
b
 f ( x)dx  F (b)  F (a)
net area
but
a
b
 | f ( x) | dx 
total area
a
If the function is non-negative, net area = area.
If the function has negative values, the integral must be split
into separate parts determined by f(x) = 0.
Integrate one part where f(x) > 0 and the other where f(x) < 0.
Indefinite Integrals and Net Change
• The integral of a rate of change is the net change.
t2
 v(t )dt  s(t ) s(t )
2
1
net change or displacement
t1
t2
 | v(t ) | dt 
total distance travelled
t1
If the function is non-negative, displacement = distance.
If the function has negative values, the integral
Must be split into separate parts determined by v = 0.
Integrate one part where v>0 and the other where v<0.
5.5 Review of Chain rule
d 
 3x 2  x  1
dx 




1
3





1
 3x 2  x  1
3

d
sin 3 
dx

2
3
 6 x  1

6x  1


3 3x 2  x  1

d
3
2 d
 sin    3 sin    sin  
dx
dx
 3sin 2  cos
2
3
If F is the antiderivative of f

f ( g ( x)) g ( x)dx  F ( g ( x))  C
u  g ( x)
du  g ( x)dx

f
(
g
(
x
))
g
(
x
)
d
x

f
(
u
)
du

F
(
u
)

C


F ( g ( x))  C

6x  1


3 3x 2  x  1
2
3
u  3x  x  1
du  (6 x  1)dx
1
 u
3
2
3 du

2
3
dx
Let u = inside function of more
complicated factor.
2


1
  (6 x  1) 3 x 2  x  1
3
dx

1

3
1
3u 3
C
1
2
 (3x  x  1) 3
Check by differentiation
C
 3sin
2
 cos d
u  sin 
du  cos d
 3 u 2du
Let u = inside function of more
complicated factor.
 u3  C
Check by differentiation
 sin3   C
Substitution with definite integrals
7

4  3xdx
0
u  4  3x
du  3dx
du
 dx
3
1
3
3 7
3
3

1 2
1 2 2
2
2
234
2
2
2
u du   u  C  (4  3x)   (25  4 ) 

3
3 3
9
9
0 9
Using a change in limits
25
3
3
1
2 
2
234
2
2
u du  u   (25  4 ) 

34
9 4 9
9
25
1
2
3
2
The average value of a function on [a, b]
b
1
Average 
f
(
x
)
dx

baa