Chapter 5 – The Definite Integral
5.1 Estimating with Finite Sums
Example Finding Distance
Traveled when Velocity Varies
A p article starts at x  0 an d m o ves alo n g th e x -ax is w ith velo city v ( t )  t
2
fo r tim e t  0 . W h ere is th e p article at t  3?
G raph v and partition the tim e interval into subintervals of length  t . If you us e
 t  1 / 4, you w ill have 12 subintervals. T he area of each rectangle approxim ates
the distance traveled over the subint erval. A dding all of the areas (distanc es)
gives an approxim ation to the total area under the curve (total distance travele d)
from t  0 to t  3.
C ontinuing in this m anner, derive the ar ea  1 / 4   m
 for each subinterval and
2
i
add them :
1

256
9
256

25
256

49
256

81
256

121
256

169
256

225
256

289
256

361
256

441
256

529
256

2300
256
 8.98
LRAM, MRAM, and RRAM approximations to the
area under the graph of y=x2 from x=0 to x=3
p.270 (1-19, 26, 27)
5.2 Definite Integrals
Sigma notation enables us to express a large sum in compact form:
Ex)
Ex)

5

2

n
k 1
a k  a1  a 2  a 3  ...  a n 1  a n
Ex)
k
k 1
k 1
k
k 1
Ex)


3
k 1
5
k 4
  1
k
2
k 1
k
k
The Definite Integral as a Limit
of Riemann Sums
L et f be a function defined on a closed i nterval [ a , b ]. For any partition P
of [ a , b ], let the num bers c be chosen arbitr arily in the subinterval [ x , x ].
k
k -1
k
n
If there exists a num ber I such that lim  f ( c )  x  I
P 0
k 1
k
k
no m atter how P and the c 's are chosen, th en f is in tegrab le on [ a , b ] and
k
I is the d efin ite in tegral of f over [ a , b ].
A ll continuous functions are integrable. T hat is, if a function f is
continuous on an interval [ a , b ], then its definite integral over
[ a , b ] exists.
b
n
We have that lim
n 
 f  c  x
k
k 1
Upper limit
Integral sign
k

 f  x  dx
a
b
x
dx



f
a
Lower limit
Variable of Integration
Integrand
Example Using the Notation
T he interval [-2, 4] is partitioned into n subintervals of equal length  x  6 / n .
Let m denote the m idpoint of the k
k
lim   3  m
n
n 
k 1
th
subinte rval. E xpress the lim it
  2 m  5  x as an integral.
2
k
k
Area Under a Curve
If y  f ( x ) is nonnegative and integrable over a closed interval [ a , b ],
then the area under the curve y  f ( x ) from a to b is the in tegral
of f from a to b , A   f ( x ) dx .
b
a
Notes about Area
A rea=   f ( x ) dx w hen f ( x )  0.
b
a
 f ( x ) dx   area above the x -axis    area below the x -ax is  .
b
a
The Integral of a Constant
If f ( x )  c , w h ere c is a co n stan t, o n th e in te rval [ a , b ], th en
 f ( x ) d x   cd x  c ( b  a )
b
b
a
a
E valuate num erically.
2
 x sin xdx
-1
FN IN T ( x sin x , x , -1, 2)  2.04
Evaluate the following integrals:
2

2
2
4  x dx
2

1
x
x
dx
p.282 (1-27, 33-39) odd
5.3 Definite
Integrals and
Antiderivatives
S uppose  f ( x ) dx  5,
1
-1
 f ( x ) dx   2,
and  h ( x ) dx  7 .
 f ( x ) dx   2,
and  h ( x ) dx  7 .
 f ( x ) dx   2,
and  h ( x ) dx  7 .
4
1
1
-1
1
Find  f ( x ) dx if possible.
4
S uppose  f ( x ) dx  5,
1
-1
4
1
1
-1
4
Find  f ( x ) dx if possible.
1
S uppose  f ( x ) dx  5,
1
-1
2
Find  h ( x ) dx if possible.
2
4
1
1
-1
1
Ex: Show that the value of 
1  cos x dx 
0
3
2
Average (Mean) Value
If f is integrable on [ a , b ], its average (m ean) value on [ a , b ] is
avg ( f ) 
1
ba
b
 f ( x ) dx
a
Find the average value of f ( x )  2  x on [0,4].
2
The Mean Value Theorem for Definite Integrals
If f is continuous on [ a , b ], then at som e p oint c in [ a , b ],
f (c ) 
1
ba
b
 f ( x ) dx .
a
Integral Formulas
x 1
n
x
n
dx 
n 1
 C , n  1
 csc
2
xdx   cot x  C
 dx   1dx  x  C
 cos kxdx 
 sin kxdx 
 sec
2
sin kx
 sec x tan xdx  sec x  C
C
k
 cos kx
C
 csc x cot xdx   csc x  C
k
xdx  tan x  C
This is known as the indefinite integral. C is a constant.
Evaluate:

5
x dx
 sin 2 xdx

1
 cos
x
dx
x
2
dx
p. 290 (1 – 29) odd
19 – 29 note
Do (31-35)
After 5.4
5.4 Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus – Part 1
If f is continuous on [ a , b ], then the function F ( x )   f ( t ) dt
x
a
has a derivative at every point x in [ a , b ], and
dF
dt

d
dx
 f ( t ) dt  f ( x ).
x
a
Evaluate the following:
x
d
dx
Find
 cos tdt

x
dx
x
2
 cos tdt
1
1

dx 1  t
0
dy
y
d
2
dt
Find
dy
dx
x
5
y 
y
 3t sin tdt
2

2x
x
Find a function y = f(x) with derivative
dy
 tan x
dx
That satisfies the condition f(3) = 5.
1
2e
t
dt
The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of [ a , b ], and if F is any antiderivative
of f on [ a , b ], then  f ( x ) dx  F ( b ) - F ( a ).
b
a
T his part of the Fundam ental T heorem is also called the In tegral
E valu ation T h eorem .
E valuate   3 x  1  dx using an antiderivative.
3
-1
2
How to Find Total Area Analytically
T o find the area betw een the graph of y  f ( x ) and the x -axis over the interval
[ a , b ] analytically,
1. partition [ a , b ] w ith the zeros of f ,
2. integrate f over each subinterval,
3. add the absolute values o f the integrals.
Find the area of the region between the curve y = 4 – x2, [0, 3] and the x-axis.
Look at page 301 example 8.
p.302 (1-57) odd
5.5 Trapezoidal Rule
 f ( x)dx 
h
b
a
y  y
0
 h
1
y  y
1
2
2
 ...  h 
h
2
n 1

y 

2 
n
 y  2 y  2 y  ...  2 y  y  ,
w h ere y  f ( a ),
0
2
n
2
0
1
 y
n 1
2
 y
 h
 y  y  ...  y
 2

y
0
1
y  f ( x ), ..., y
1
n 1
2
1
n 1
n
 f ( x ), y  f ( b ).
n 1
n
The Trapezoidal Rule
b
T o approxim ate  f ( x ) dx ,
use
a
T 
h
2
 y  2 y  2 y  ...  2 y  y  ,
0
1
n 1
2
n
w here [ a , b ] is partitioned into n subinter vals of equal length
h  (b - a ) / n.
E quivalently, T 
LR AM  R R AM
n
n
,
2
w here L R A M and R R A M are the R ienam m sum s using the left
n
n
and right endpoints, respectively, for f for the partition.
2
Use the trapezoidal rule with n = 4 to estimate

2
x dx . Compare with fnint.
1
Ex: An observer measures the outside temperature every hour from noon until
midnight, recording the temperatures in the following table.
Time
N
1
2
3
4
5
6
7
8
9
10
11
M
Temp
63
65
66
68
70
69
68
68
65
64
62
58
55
What was the average temperature for the 12-hour period?
Simpson’s Rule
b
T o ap p ro x im ate  f ( x ) d x , u se
a
S 
h
3
 y  4 y  2 y  4 y  ...  2 y
0
1
2
3
 4y
n2
n 1
 y
n
,
w h ere [ a , b ] is p artitio n ed in to an even n u m b er n su b in tervals
o f eq u al len g th h  ( b - a ) / n .
2
Ex: Use Simpson’s rule with n = 4 to approximate
 5 x dx
4
0
p.312 (1-18)