Chapter 5 – The Definite Integral
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Transcript Chapter 5 – The Definite Integral
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
ba
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
ba
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
2e
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
n2
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)