The Precise Definition of the Definite
Integral
Introduction
The Area Under a Curve
The area A of the region that lies under the graph of the continuous function f is the limit
of the sum of the areas of approximating rectangles
A = lim [f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x]
n→∞
Sigma Notation
We often use sigma notation to write sums with many terms more compactly. For instance:
n
X
f (x∗i )∆x = f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x
i=1
Example 1.
Express the following sums using sigma notation:
• 1+2+3+4+5+6+7+8
• 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82
• 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83
The Precise Definition of the Definite Integral
Properties of Sigma Notation
•
n
X
c = nc
i=1
•
n
X
cai = c
i=1
•
•
n
X
n
X
!
ai
i=1
(ai + bi ) =
n
X
i=1
i=1
n
X
n
X
(ai − bi ) =
i=1
!
ai
+
n
X
!
bi
i=1
!
ai
−
i=1
n
X
!
bi
i=1
Three Useful Sums
•
n
X
i=
i=1
•
n
X
n(n + 1)
2
i2 =
i=1
•
n
X
i3 =
i=1
n(n + 1)(2n + 1)
6
n(n + 1)
2
2
Example 2.
Find the following sums:
• 1+2+3+4+5+6+7+8
• 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82
• 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83
The Precise Definition of the Definite Integral
The Definite Integral
The Area Under a Curve
The area A of the region that lies under the graph of the continuous function f is the limit
of the sum of the areas of approximating rectangles
A = lim [f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x]
n→∞
The Definite Integral
Let f be a function defined for a ≤ x ≤ b. The definite integral of f from a to b is:
ˆ
b
f (x) dx = lim
a
n→∞
n
X
f (x∗i )∆x
i=1
provided that this limit exists and gives the same value for all possible choices of sample points.
If it does exist, we say that f is integrable on [a, b].
What Functions are Integrable?
If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is
´b
integrable on [a, b]; that is, the definite integral a f (x) dx exists.
The Precise Definition of the Definite Integral
How to Compute the Definite Integral of an Integrable Function
Using Right Endpoints
If f is integrable on [a, b], then
ˆ
b
f (x) dx = lim
n→∞
a
where
∆x =
b−a
n
and
n
X
f (xi )∆x
i=1
xi = a + i∆x
Example 3.
Express the following limit as an integral on the interval [0, π]:
lim
n→∞
n
X
i=1
x3i + xi sin xi ∆x
The Precise Definition of the Definite Integral
Example 4.
Evaluate the Riemann sum for f (x) = x3 − 6x, taking the sample points to be right endpoints and
a = 0, b = 3, and n = 6.
-
The Precise Definition of the Definite Integral
Example 5.
ˆ 3
x3 − 6x dx.
Evaluate
0
-
The Precise Definition of the Definite Integral
The Midpoint Rule
The Midpoint Rule
ˆ
b
f (x) dx ≈
a
where
n
X
f (xi )∆x = [f (x1 ) + f (x2 ) + · · · + f (xn )] ∆x
i=1
b−a
∆x =
n
1
xi = a + i −
∆x
2
and
Example 6.
ˆ
2
Use the Midpoint Rule with n = 5 to approximate
1
1
dx.
x