Announcements
Topics:
- sections 7.3 (definite integrals) and 7.4 (FTC)
* Read these sections and study solved examples in your
textbook!
Work On:
- Practice problems from the textbook and assignments
from the coursepack as assigned on the course web
page (under the link “SCHEDULE + HOMEWORK”)
Area
How do we calculate the
area of some irregular
shape?
For example, how do we
calculate the area under
the graph of f on [a,b]?
Area  ?

Area
Approach:
We approximate the area using rectangles.
number of
rectangles:
n4
width of each
rectangle:
ba
x 
n

x0 
x1
x2
x3
 x4
Area
Left-hand estimate:
Let the height of each rectangle be given by the value
of the function at the left endpoint of the interval.
x0 
x1
x2
x3
 x4


Area
Left-hand estimate:
Area  f (x0 )x  f (x1)x  f (x2 )x  f (x3 )x
 ( f (x0 )  f (x1)  f (x2 )  f (x3 ))x
3
  f (x i )x
i 0
Riemann Sum
Area
Right-hand estimate:
Let the height of each rectangle be given by the value
of the function at the right endpoint of the interval.
x0 
x1
x2
x3
 x4


Area
Right-hand estimate:
Area  f (x1)x  f (x2 )x  f (x3 )x  f (x4 )x
 ( f (x1)  f (x2 )  f (x3 )  f (x4 ))x
4
  f (x i )x
i1
Riemann Sum
Area
Midpoint estimate:
Let the height of each rectangle be given by the value
of the function at the midpoint of the interval.
x1
x2
x3
x4
Area
Midpoint estimate:
Area  f (x )x  f (x )x  f (x )x  f (x )x
*
1
*
2
*
3
*
4
 ( f (x )  f (x )  f (x )  f (x ))x
*
1
*
2
*
3
*
4
4
  f (x )x
*
i


i1
Riemann Sum
Area
How can we improve our estimation?
Increase the number of rectangles!!!
16
Area   f (t )t
*
i
i1
How do we make it exact?

Let the number of rectangles go to infinity!!!
Area
How can we improve our estimation?
Increase the number of rectangles!!!
16
Area   f (x )x
*
i
i1
How do we make it exact?

Let the number of rectangles go to infinity!!!
Area
How can we improve our estimation?
Increase the number of rectangles!!!
16
Area   f (x )x
*
i
i1
How do we make it exact?

Area
How can we improve our estimation?
Increase the number of rectangles!!!
16
Area   f (x )x
*
i
i1
How do we make it exact?

Let the number of rectangles go to infinity!!!
Riemann Sums and the
Definite Integral
Definition:
The definite integral of a function f on the
interval from a to b is defined as a limit of the
Riemann sum
b

n
*

f (x)dx  lim  f (x i )x
n 
a
*
i
i1
where x is some sample point in the interval
[x i1, x i ] and x  b  a .

n
The Definite Integral
Interpretation:
If f  0 , then the definite
integral is the area under
the curve y  f (x) from
a to b.
Area 
b
 f (x)dx
a


Estimating a Definite Integral
3
Estimate
 ln xdx
using left-endpoints,
1
midpoints, and right-endpoints with n=4.

The Definite Integral
Interpretation:
If f is both positive and
negative, then the definite
integral represents the
NET or SIGNED area, i.e.
the area above the x-axis
and below the graph of f
minus the area below the
x-axis and above f
4

1
f (x)dx  net area
Evaluating Definite Integrals
Example:
Evaluate the following integrals by interpreting
each in terms of area.
3
1
(a)

1  x 2 dx
(b)
0
0
(c)


 sin x dx

 (x 1) dx

Properties of Integrals
Assume that f(x) and g(x) are continuous
functions and a, b, and c are real numbers such
that a<b.
a
(1)
 f (x) dx  0
a
(2)
(3)
b
a
a
b
 f (x) dx    f (x) dx
b
b
a
a
 c f (x) dx  c  f (x) dx
Properties of Integrals
Assume that f(x) and g(x) are continuous
functions and a, b, and c are real numbers such
that a<b.
(4)
b
b
b
a
a
a
  f (x) g(x)dx   f (x) dx   g(x) dx
b
(5)
 c dx  c(b  a)
a
Summation Property of the
Definite Integral
(6) Suppose f(x) is continuous on the interval
from a to b and that a  c  b.

Then
b
c
 f (x) dx
 f (x) dx
a
c
b 
c
a
a

b
 f (x) dx   f (x) dx   f (x) dx .
c
Properties of the Definite Integral
(7) Suppose f(x) is continuous on the interval
from a to b and that m  f (x)  M.

Then m(b  a) 
b
 f (x) dx  M(b  a).
a
Types of Integrals
• Indefinite Integral
function of x
 f (x) dx  F(x)  C
antiderivative of f
• Definite Integral

b

a
number
f (x) dx  net area
The Fundamental Theorem of Calculus
If f is continuous on [a, b], then
b

a
f (x) dx  F(x) ba  F(b)  F(a)

where F is any antiderivative of f , i.e., F' f .




Evaluating Definite Integrals
Example:
Evaluate each definite integral using the FTC.
3
2
(a)  (x 1) dx
(b)  ( 4  t )dt
0
1
(c) 
2
 14
1
1
1 4x
2
dx
(d)

2

1
t
4
(3x 1) 2
dx
x
Evaluating Definite Integrals
Example:
Try to evaluate the following definite integral
using the FTC. What is the problem?
4

1
1
2 dx
(x  2)
Differentiation and Integration
as Inverse Processes
If f is integrated and then differentiated, we
arrive back at the original function f.
d
dx
x
 f (t) dt  f (x)
FTC I
a
If F is differentiated and then integrated, we
arrive back at the original function F.

b

a
d
b
F(x) dx  F(x) a
dx
FTC II
The Definite Integral - Total Change
Interpretation:
The definite integral represents the total
amount of change during some period of time.
Total change in F between times a and b:
F(b)  F(a) 
value at end
value at start
b

a
dF
dt
dt
rate of change
Application – Total Change
Example:
Suppose that the growth rate of a fish is given
by the differential equation
dL
0.09t
 6.48e
dt
where t is measured in years and L is measured
in centimetres and the fish was 0.0 cm at age
t=0 (time
measured from fertilization).
Application – Total Change
(a) Determine the amount
the fish grows between 2
and 5 years of age.
(b) At approximately what
age will the fish reach
45cm?
Application – Total Change
(a) Determine the amount
the fish grows between 2
and 5 years of age.
L(5)  L(2) 

5

2
5
(b) At approximately what
age will the fish reach
45cm?
dL
dt
dt
0.09t
6.48e
dt

2
 72e

0.09t 5
2
 72e0.09(5)  72e0.09(2) 
 14.2 cm
Application – Total Change
(a) Determine the amount
the fish grows between 2
and 5 years of age.
(b) At approximately what
age will the fish reach
45cm?
dL
 dt dt
  6.48e0.09t dt
L(t) 
 72e0.09t  C

L(0)  0  C  72


L(t)  72e0.09t  72
L(t)  45 when t 11 years