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Chapter 5 – Integrals
5.2 The Definite Integral
5.2 The Definite IntegralGeorg Friedrich Bernhard Riemann
1826 - 1866
5.2 The Definite Integral2
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
Review - Riemann Sum21
18
V t
subinterval
partition
Subintervals do not all have to be the same size.
5.2 The Definite Integral3
a) If you are given the information above, evaluate the Riemann sum with n=6, taking sample points to be right endpoints. What does the Riemann sum illustrate? Illustrate with a diagram.
b) Repeat part a with midpoints as sample points.
Example 1 – pg. 382 # 4
3( ) sin 0 2f x x x
5.2 The Definite Integral4
Idea of the Definite Integral
1
limn
in
i
f x x
is called the definite integral of
over .f ,a b
If we use subintervals of equal length, then the length of a subinterval is:
i
b ax x a i x
n
The definite integral is then given by:
1
limn
in
i
f x x
5.2 The Definite Integral5
Definite Integral in Leibnitz Notation
1
limn
in
i
f x x
Leibnitz introduced a simpler notation for the definite integral:
1
limn b
i ani
f x x f x dx
Note that the very small change in x becomes
dx.
5.2 The Definite Integral6
Explanation of the Notation
b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrand
variable of integration(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
5.2 The Definite Integral7
If f is continuous on [a, b], or if f has only a
finite number of jump discontinuities, then f is
integrable on [a, b]; that is, the definite
integral exists.
Theorem (3)
( )b
af x dx
5.2 The Definite Integral8
Putting all of the ideas together, if f is differentiable on [a, b], then
where
Theorem (4)
1
limn b
i ani
f x x f x dx
i
b ax x a i x
n
5.2 The Definite Integral9
Use the midpoint rule with the given value of n to approximate the integral. Round your answers to four decimal places.
Example 2
/2 4
0cos 4xdx n
5.2 The Definite Integral10
1.
2.
3.
4.
5.
6.
7.
Evaluating Integrals using Sums
1
( 1)
2
n
i
n ni
2
1
( 1)(2 1)
6
n
i
n n ni
23
1
( 1)
2
n
i
n ni
1
n
i
c nc
1 1
n n
i ii i
ca c a
1 1 1
n n n
i i i ii i i
a b a b
1 1 1
n n n
i i i ii i i
a b a b
5.2 The Definite Integral11
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
Example 4 – Page 377 #23
0 2
2x x dx
5.2 The Definite Integral12
Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
Example 5
2 3
1x dx
5.2 The Definite Integral13
Express the integral as a limit of Riemann sums. Do not evaluate the limit.
Example 6 – Page 383 # 29
6
52 1
xdx
x
5.2 The Definite Integral14
Express the limit as a definite integral on the given interval.
Example 7 – page 383 # 17
2
1
lim ln 1 [2,6]n
i in
i
x x x
5.2 The Definite Integral15
Express the limit as a definite integral.
Example 8 – page 385 # 71
4
51
4
lim
Hint: Consider ( )
n
ni
i
n
f x x
5.2 The Definite Integral16
1.
2.
3.
4.
Properties of the Integral
( )b
acdx c b a
( ) ( ) ( ) ( )b b b
a a af x g x dx f x dx g x dx
( ) ( )b b
a acf x dx c f x dx
( ) ( ) ( ) ( )b b b
a a af x g x dx f x dx g x dx
5.2 The Definite Integral17
5.
6.
7.
8.
Properties Continued
( ) ( ) ( )c b b
a c af x dx f x dx f x dx
If ( ) 0 for then, ( ) 0b
af x a x b f x dx
If ( ) ( ) for then, ( ) ( )b b
a af x g x a x b f x dx g x dx
If ( ) for then,
( ) ( ) ( )b
a
m f x M a x b
m b a f x dx M b a
5.2 The Definite Integral18
Use Property 8 to estimate the value of the integral.
Example 9 – page 384 # 62
2 3
03 3x x dx
5.2 The Definite Integral19
Evaluate the integral by interpreting it in terms of areas.
Example 10 – page 384 # 37
0 2
31 9 x dx
5.2 The Definite Integral20
Work in groups to prove the following:
Example 11 – page 383 # 28
3 32
3
b
a
b ax dx
5.2 The Definite Integral21
We will be evaluating Leibnitz integrals using the idea of antiderivatives and the fundamental theorem of calculus.
What to expect next…
