Download Quick Review of the Definition of the Definite Integral

Quick Review of the De…nition of the De…nite Integral
Philippe B. Laval
KSU
Today
Philippe B. Laval (KSU)
De…nite Integral
Today
1 / 25
Introduction
This is a quick review of the material related to the de…nition of the
de…nite integral. The following topics are reviewed:
1
Sigma notation
2
The area problem
3
De…nition of the de…nite integral
4
The Fundamental Theorem of Calculus.
Philippe B. Laval (KSU)
De…nite Integral
Today
2 / 25
Sigma Notation
De…nition (sigma notation)
Let am ; am+1 ; :::; an be real numbers and let m and n be integers such
that m n.
n
X
1 We represent a
ai and we say ”The
+
a
+
:::
+
a
+
a
by
m
m+1
n 1
n
i =m
sum as i ranges from m to n of a sub i”.
2
3
i is called the index of summation.
n
X
ai is the sum written in ”sigma” notation.
i =m
am + am+1 + ::: + an
Philippe B. Laval (KSU)
1
+ an is the sum written in ”expanded” form.
De…nite Integral
Today
3 / 25
Sigma Notation
Remark
Let us make the following remarks.
1
When writing
n
X
ai , the index of summation takes on all integer
i =m
values between m and n:
2
The name of the index of summation is not important as it is being
replaced by integers when expanding the sum. In other words
n
X
i =m
ai =
n
X
aj
j =m
Usually, we use the letters i, j or k for index of summation.
Philippe B. Laval (KSU)
De…nite Integral
Today
4 / 25
Sigma Notation
Examples
1
2
10
X
i =1
10
X
i 2 = 12 + 22 + 32 + ::: + 102
2i = 2 + 22 + 23 + ::: + 210
i =1
3
4
5
10
X
i =1
n
X
i =0
n
X
( 1)i 2i =
2 + 22
23 + ::: + 210
xi
x2
x3
xn
=1+x +
+
+ ::: +
i!
2!
3!
n!
f (xi ) = f (x1 ) + f (x2 ) + f (x3 ) + ::: + f (xn )
i =1
Philippe B. Laval (KSU)
De…nite Integral
Today
5 / 25
The Area Problem
The goal is to …nd the area A of the region bounded by the graph of
y = f (x), the x-axis, the vertical lines x = a and x = b, that is the area of
the shaded region below.
Figure: Area below a graph
Philippe B. Laval (KSU)
De…nite Integral
Today
6 / 25
The Area Problem: Approximation of the Area
Divide [a; b] into n subintervals of equal
length: [x0 ; x1 ], [x1 ; x2 ], :::, [xn 1 ; xn ]
In each subinterval [xi
point we call xi
Figure: Approximating an area
with rectangles
1 ; xi ]
pick a
For each subinterval [xi 1 ; xi ] form a
rectangle Ri of width
b a
x = xi xi 1 =
and height
n
f (xi )
b a
The area of each Ri is Ai =
f (xi )
n
n
n
X
X
b a
A
Ai =
f (xi ) =
n
i =1
i =1
n
b aX
f (xi )
n
i =1
Philippe B. Laval (KSU)
De…nite Integral
Today
7 / 25
The Area Problem
Remarks and example
In theory, xi can be selected anywhere in [xi 1 ; xi ]. In practice, it will
usually be one of the end points, or the midpoint, or the point at
which f is either maximum or minimum.
n
b aX
The sum
f (xi ) is called a Riemann sum. If xi
n
i =1
corresponds to a maximum of f in [xi 1 ; xi ] for each i = 1; 2; :::; n
then the sum is called an upper Riemann sum. If xi corresponds to
a minimum of f then the sum is called a lower Riemann sum.
Example
Approximate the area below the graph of y = x 2 between x = 0 and x = 4,
using 4 subintervals and by selecting xi to be the right end point of each
subinterval. Repeat the procedure by selecting xi to be the left end point.
Philippe B. Laval (KSU)
De…nite Integral
Today
8 / 25
The Area Problem
More remarks
You can practice with the applet at Riemann Sums
The value we get for the area is always larger than the lower Riemann
sum and smaller than the upper Riemann sum (why?).
As n gets larger, the approximation of the area gets better and better.
This suggest that we can …nd the exact value of the area by letting n
go to in…nity. In other words, we de…ne the area below the graph of
y = f (x) between x = a and x = b by
!
n
n
X
b aX
A = lim
f (xi ) x = lim
f (xi )
n!1
n!1
n
i =1
i =1
Philippe B. Laval (KSU)
De…nite Integral
Today
9 / 25
The De…nite Integral: De…nition
De…nition (de…nite integral)
The de…nite integral of f from a to b is de…ned by
Zb
a
f (x) dx = lim
n!1
n
X
f (xi ) x
i =1
If the limit exists, f is said to be integrable. a is called the lower limit of
integration, b the upper limit of integration and f the integrand.
Remark
What functions are integrable. The answer is not simple. You will study
this question in a more advanced class such as Real Analysis. For now,
know that if f is continuous then f is integrable. Keep in mind that this
does not mean if f is not continuous it is not integrable.
Philippe B. Laval (KSU)
De…nite Integral
Today
10 / 25
The De…nite Integral: Relationship With Area
Proposition
The relation between an integral and the area of the region bounded by
y = f (x), the x-axis, the lines x = a and x = b is as follows:
1
If f (x)
0 then
Rb
f (x) dx is the area of the region between the
a
2
graph of f and the x-axis and between x = a and x = b.
Rb
If f (x) 0 then f (x) dx is the negative of the area of the region
a
3
between the graph of f and the x-axis and between x = a and x = b.
Rb
In general, f (x) dx = A1 A2 where A1 is the area of the region
a
above the x-axis, below the graph of f and A2 is the area of the
region below the x-axis, above the graph of f .
Philippe B. Laval (KSU)
De…nite Integral
Today
11 / 25
The De…nite Integral: Relationship With Area
R1
0
R3
f (x) dx = A1
f (x) dx =
A2 (remember, the
1
numbers Ai represent areas, they are
positive).
R5
f (x) dx =
3
R3
Figure: Relationship between
integral and area
0
R5
1
R5
f (x) dx =
f (x) dx =
f (x) dx =
0
Philippe B. Laval (KSU)
De…nite Integral
Today
12 / 25
The De…nite Integral: Relationship With Area
If on the other hand we could compute
Rb
f (x) dx for any a and b, then we could use
a
this knowledge to compute areas as follows:
A1 =
A2 =
A3 =
area of the shaded region:
Figure: Relationship between
integral and area
Philippe B. Laval (KSU)
De…nite Integral
Today
13 / 25
The De…nite Integral: Properties
Rb
a
Ra
a
Rb
f (x) dx =
Ra
f (x) dx
Rb
b
f (x) dx = 0
cdx = c (b
a
Rb
[f (x)
g (x)] dx =
f (x) dx =
f (x) dx
Rc
f (x) dx +
a
f (x) dx
Rb
Rb
f (x) dx
c
g (x) dx
a
a
a
If f (x)
Rb
Rb
a
a
a)
a
Rb
cf (x) dx = c
0 for x in [a; b] then
Rb
f (x) dx
0
a
If f (x)
g (x) for x in [a; b] then
Rb
f (x) dx
a
Rb
g (x) dx
a
If f has a maximum value M on [a; b] and a minimum value m on
Rb
[a; b] then m (b a)
f (x) dx M (b a)
a
Philippe B. Laval (KSU)
De…nite Integral
Today
14 / 25
The De…nite Integral: Computation
For now, we have two ways to compute integrals: approximate them using
Riemann sums, or compute them using the area interpretation of integrals.
Both methods are very limited, we illustrate them with a few examples.
Example
Approximate
R4
x 2 dx using Riemann sums. You will use 4 subintervals, and
0
select xi to be the right end point of each subinterval.
Example
Compute
R2
xdx using the area interpretation of the integral.
0
Example
Compute
R2 p
4
x 2 dx using the area interpretation of the integral.
0
Philippe B. Laval (KSU)
De…nite Integral
Today
15 / 25
The De…nite Integral: Antiderivatives
De…nition
An antiderivative of a function f is a function F such that F 0 (x) = f (x).
Example
Since the derivative of ln x is
1
1
, it means that an antiderivative of is ln x
x
x
Example
Since (sin x)0 = cos x, it means that an antiderivative of cos x is sin x.
Remark
Antiderivatives are not unique. If F is an antiderivative of f (that is if
F 0 = f ), then F + C where C is any constant is also an antiderivative of f .
Philippe B. Laval (KSU)
De…nite Integral
Today
16 / 25
The Fundamental Theorem of Calculus
Theorem (Fundamental Theorem of Calculus)
Let f be a continuous function on [a; b].
1
The function g (x) =
Rx
f (t) dt for x in [a; b] is continuous and
a
0
di¤erentiable. Furthermore, g (x) = f (x)
that is g is an antiderivative of f .
2
or
d
dx
Rx
f (t) dt = f (x)
a
If F is any antiderivative of f then
Rb
f (x) dx = F (x)jba = F (b) F (a).
a
Philippe B. Laval (KSU)
De…nite Integral
Today
17 / 25
The Fundamental Theorem of Calculus
Remark
The following follows from the theorem:
1
Part 1 of the Fundamental Theorem of Calculus says that the
derivative of the integral of a function is the function itself.
2
Part 2 of the Fundamental Theorem of Calculus provides a way to
compute integrals. We simply have to …nd an antiderivative of the
integrand, and plug in the limits of integration.
De…nition (inde…nite integral)
R
f (x) dx is used to represent an antiderivative of f . That is
Z
f (x) dx = F (x) , F 0 (x) = f (x)
It is a function, not a number. It is called the inde…nite integral of f .
Philippe B. Laval (KSU)
De…nite Integral
Today
18 / 25
The Fundamental Theorem of Calculus
Remark
Let us make a couple of remarks:
1
2
above de…nition, if we replace f (x) by F 0 (x), we obtain
RIn the
0
F (x) dx = F (x). In other words, the integral of the derivative of a
function is the function itself. If we combine this with part 1 of the
Fundamental Theorem of Calculus which says that the derivative of
the integral of a function is the function itself, we see that integration
and di¤erentiation are inverse processes. One undoes what the other
one does.
If we combine part 1 of the Fundamental Theorem of Calculus with
the chain rule, then we have with u being a function of x:
d
dx
Zu
f (t) dt = f (u)
du
= f (u) u 0 (x)
dx
a
Philippe B. Laval (KSU)
De…nite Integral
Today
19 / 25
The Fundamental Theorem of Calculus
At this point, you should know the following antiderivatives:
R
R n
u n+1
sec2 udu = tan u + C
+ C if n 6= 1
u du =
R
n+1
csc2 udu = cot u + C
R 1
R
du = ln juj + C
sec u tan udu = sec u + C
R uu
R
u
e du = e + C
csc u cot udu = csc u + C
u
R u
R
a
1
a du =
+C
du = tan 1 u + C
2 +1
ln
a
u
R
R
1
sin udu = cos u + C
p
du = sin 1 u + C
R
2
1 u
cos udu = sin u + C
Philippe B. Laval (KSU)
De…nite Integral
Today
20 / 25
The Fundamental Theorem of Calculus: Examples
Example
Find
R3
x 2 dx
1
Example
Find
Z
sin xdx
0
Example
Find
Z2
x + x 2 dx
0
Philippe B. Laval (KSU)
De…nite Integral
Today
21 / 25
The Fundamental Theorem of Calculus: Examples
Example
Find
Z2
x2 + x
2
dx
0
Example
Find
R3
x 2 dt
1
Example
Find
Z
sin xdt
0
Philippe B. Laval (KSU)
De…nite Integral
Today
22 / 25
The Fundamental Theorem of Calculus: Examples
Example
0 x
1
Z
d @
Find
sin tdt A
dx
0
Example
0 0
1
Z
d @
Find
sin tdt A
dx
x
Philippe B. Laval (KSU)
De…nite Integral
Today
23 / 25
The Fundamental Theorem of Calculus: Examples
Example
Find
0
d B
@
dx
Zx 2
1
1
C
sec tdt A
Example
0 2
1
Zx
d B
C
Find
@ sin tdt A
dx
x
Philippe B. Laval (KSU)
De…nite Integral
Today
24 / 25
Exercises
See the problems at the end of my notes on the de…nite integral. These
notes contain four sections. There is a set of problems at the end of each
section.
Philippe B. Laval (KSU)
De…nite Integral
Today
25 / 25