MTH132
Chapter 4 - Integrals
Michigan State University
Appendix E - Sigma Notation
Definition(s) 0.1. If am , am+1 , . . . , an−1 , an are real numbers and m and n are integers such that m ≤ n, then
n
X
ai = am + am+1 + · · · + an−1 + an
i=m
Definition(s) 0.2. This way of short-handing sums of many numbers is called sigma notation (uses the Greek letter Σ
“Sigma”). The letter i above is called the index of summation and it takes on consecutive integer values starting with
m and ending with n.
Theorem 0.4. Let c be a constant and n a positive
integer. Then
Theorem 0.3. If c is any constant then:
n
X
1.
i=m
n
X
2.
n
X
ai
1.
i=m
(ai + bi ) =
n
X
ai +
n
X
i=m
i=m
n
X
n
X
n
X
i=m
(ai − bi ) =
n
X
1=n
i=1
i=m
3.
2
cai = c
ai −
i=m
bi
2.
n
X
i=
i=1
bi
3.
i=m
n
X
n(n + 1)
2
i2 =
i=1
n(n + 1)(2n + 1)
6
The Definite Integral
a
a
b
b
a
Theorem 2.1 (Definite Integral). If f is integrable on [a, b], then
Z
b
f (x) dx = lim
n→∞
a
where
∆x =
Z
Remark 2.2. The definite integral,
b−a
n
and
n
X
f (xi )∆x
i=1
xi = a + i∆x
b
f (x) dx gives the net area between the curve f and the x− axis.
a
1
b
MTH132
Chapter 4 - Integrals
Michigan State University
Example 2.3 (Instructor). Evaluate the following sums
1.
4
X
2k − 1
2k + 1
i=0
2.
4
X
(2 − 3i)
i=0
3.
38
X
(3i − 3i−1 )
i=1
√
Example 2.4 (Instructor). Write th sum:
3+
√
4 + ··· +
√
25
in sigma notation
Example 2.5 (Instructor). Use the definition of the definite
Z 3
x+1
integral to verify that
dx = 3
2
1
1
3
Z
x2 + 1 dx
Example 2.6 (Instructor). Evaluate the definite integral
1
Example 2.7 (Student). Evaluate the following sums
1.
3
X
23−i
i=−1
2.
4
X
(1 − i)(2 + i)
i=0
Example 2.8 (Student). Write th sum:
√
3−
√
5+
√
7−
Z 4
Example 2.9 (Student). Evaluate the definite integral
√
9 + ··· +
√
27
in sigma notation
x2 − x + 1 dx
−1
Z
Example 2.10 (Student). Evaluate the definite integral
7
f (x) dx.
1
Where f (x) is given by the function to the right.
Z
Example 2.11 (Student, Bonus: as time allows). Prove that
a
2
b
x2 dx =
b3 − a3
3
3