5 Integration
5.1 Sums and Sigma Notation
De
nition 1 Sigma Notation
If m and n are integers with m ≤ n, and if f is a function de
ned at the integers m, m + 1, m +
2, ..., n, the symbol ni=m f (i) represents the sum of the values of f at those integers:
P
n
X
f (i) = f (m) + f (m + 1) + f (m + 2) + · · · + f (n).
i=m
The explicit sum appearing pn the right side of the equation is the expansion of the sum
represented in sigma notation on the left side.
Note i is the index of summation, use i = j + m for all i. The index of summation is a
dummy variable. The limits of summation: m is the lower limit, and n is the upper limit.
Theorem 5.1 Summation Formulas (Closed Form)
n
X
1 = 1 + 1 + 1 + · · · + 1 = n, (n terms)
i=1
n
X n(n + 1)
i = 1 + 2 + 3 + ··· + n =
i=1
2
n
X n(n + 1)(2n + 1)
i2 = 12 + 22 + 32 + · · · + n2 =
i=1
6
n
X rn − 1
ri−1 = 1 + r + r2 + r3 + · · · + rn−1 =
i=1
r−1
A sum of the form ni=m (f (i + 1) − f (i)) telescopes out to the closed form f (n + 1) − f (m)
P
because all but the
rst and last terms cancel out, this is called a telescoping sum.
5.2 Areas as Limits of Sums
The Basic Area Problem
Divide [a, b] into n subintervals:
a = x0 < x1 < x2 < · · · < xn = b.
Denote by ∆xi the length of the ith subinterval [xi−1 , xi ]:
∆xi = xi − xi−1 , (i = 1, 2, 3, ..., n).
Then build a rectangle with width ∆xi and height f (xi ). The sum of these areas is given by:
n
X
Sn = f (x1 )∆x1 + f (x2 )∆x2 + f (x3 )∆x3 + · · · + f (xn )∆xn = f (xi )∆xi .
i=1
1
, Thus, Area of R = limn→∞ Sn , where max ∆xi → 0.
For equal subinterval lengths,
b−a i
∆xi = ∆x = , xi = a + i∆x = a + (b − a).
n n
5.3 The De
nite Integral
Let P be a
nite set of point arranged in order from a to b on the real line, thus
P = {x0 , x1 , x2 , ..., xn },
is called a partition of [a, b].
n depends on the partition, so n = n(P ), with length ∆xi , (f or 1 ≤ i ≤ n), where the greatest
of these numbers is the norm of P , denoted:
kP k = max ∆xi .
De
nition 2 Upper and Lower Riemann Sums
The lower Riemann sum, L(f, P ), and the upper Riemann sum, U (f, P ), for the function
f and the partition P are de
ned by:
n
X
L(f, P ) = f (l1 )∆x1 + · · · + f (ln )∆xn = f (li )∆xi ,
i=1
n
X
U (f, P ) = f (u1 )∆x1 + · · · + f (un )∆xn = f (ui )∆xi .
i=1
De
nition 3 The De
nite Integral
Suppose there is exactly one number I such that for every partition P of [a, b] we have
L(f, P ) ≤ I ≤ U (f, P ).
Then we say that the function f is integrable on [a, b], and we call I the de
nite integral
of f on [a, b]. The de
nite integral is denoted by the symbol
Z b
I= f (x)dx.
a
The dummy variable of the de
nite integral is x.
For all partitions P of [a, b], we have
Z b
L(f, P ) ≤ f (x)dx ≤ U (f, P )
a
Given a partition P having kP k = max ∆xi , chose a point ci (called a tag ) in each subinterval
and let c = (c1 , c2 , ..., cn ) denote the set of these tags. The sum
n
X
R(f, P, c) = f (ci )∆xi = f (c1 )∆x1 + · · · + f (cn )∆xn
i=1
2
, is called the Riemann sum of f on [a, b] corresponding to partition P and tags c.
The limit of a Riemann sum is the de
nite integral, that is
Z b
lim R(f, P, c) = f (x)dx
n(P )→∞, kP k→0 a
Theorem 5.2 If f is continuous on [a, b], then f is integrable on [a, b].
It is su
cient that, for any given , we should be able to
nd a partition P of [a, b] for which
U (f, P ) − L(f, P ) < , this restricts there to be only one I .
5.4 Properties of the De
nite Integral
If a > b, we have ∆xi < 0 for each i, so the integral will be negative for positive functions f and
vise versa.
Theorem 5.3 Properties of the De
nite Integral
Let f and g be integrable on an interval containing the points a, b, and c. Then
(a) An integral over an interval of zero length is zero.
Z a
f (x)dx = 0.
a
(b) Reversing the limits of integration changes the sign of the integral.
Z a Z b
f (x)dx = − f (x)dx.
b a
(c) An integral depends linearly on the integrand. If A and B are constants, then
Z b Z b Z b
(Af (x) + Bg(x))dx = A f (x)dx + B g(x)dx.
a a a
(d) An integral depends additively on the interval of integration.
Z b Z c Z c
f (x)dx + f (x)dx = f (x)dx.
a b a
(e) If a ≤ b and f (x) ≤ g(x) for a ≤ x ≤ b, then
Z b Z b
f (x)dx ≤ g(x)dx.
a a
(f) The triangle inequality for sums extends to de
nite integrals. If a ≤ b, then
Z b Z b
f (x)dx ≤ |f (x)|dx.
a a
(g) The integral of an odd function over an interval symmetric about zero is zero. If f is an
odd function, then Z a
f (x)dx = 0.
−a
3
5.1 Sums and Sigma Notation
De
nition 1 Sigma Notation
If m and n are integers with m ≤ n, and if f is a function de
ned at the integers m, m + 1, m +
2, ..., n, the symbol ni=m f (i) represents the sum of the values of f at those integers:
P
n
X
f (i) = f (m) + f (m + 1) + f (m + 2) + · · · + f (n).
i=m
The explicit sum appearing pn the right side of the equation is the expansion of the sum
represented in sigma notation on the left side.
Note i is the index of summation, use i = j + m for all i. The index of summation is a
dummy variable. The limits of summation: m is the lower limit, and n is the upper limit.
Theorem 5.1 Summation Formulas (Closed Form)
n
X
1 = 1 + 1 + 1 + · · · + 1 = n, (n terms)
i=1
n
X n(n + 1)
i = 1 + 2 + 3 + ··· + n =
i=1
2
n
X n(n + 1)(2n + 1)
i2 = 12 + 22 + 32 + · · · + n2 =
i=1
6
n
X rn − 1
ri−1 = 1 + r + r2 + r3 + · · · + rn−1 =
i=1
r−1
A sum of the form ni=m (f (i + 1) − f (i)) telescopes out to the closed form f (n + 1) − f (m)
P
because all but the
rst and last terms cancel out, this is called a telescoping sum.
5.2 Areas as Limits of Sums
The Basic Area Problem
Divide [a, b] into n subintervals:
a = x0 < x1 < x2 < · · · < xn = b.
Denote by ∆xi the length of the ith subinterval [xi−1 , xi ]:
∆xi = xi − xi−1 , (i = 1, 2, 3, ..., n).
Then build a rectangle with width ∆xi and height f (xi ). The sum of these areas is given by:
n
X
Sn = f (x1 )∆x1 + f (x2 )∆x2 + f (x3 )∆x3 + · · · + f (xn )∆xn = f (xi )∆xi .
i=1
1
, Thus, Area of R = limn→∞ Sn , where max ∆xi → 0.
For equal subinterval lengths,
b−a i
∆xi = ∆x = , xi = a + i∆x = a + (b − a).
n n
5.3 The De
nite Integral
Let P be a
nite set of point arranged in order from a to b on the real line, thus
P = {x0 , x1 , x2 , ..., xn },
is called a partition of [a, b].
n depends on the partition, so n = n(P ), with length ∆xi , (f or 1 ≤ i ≤ n), where the greatest
of these numbers is the norm of P , denoted:
kP k = max ∆xi .
De
nition 2 Upper and Lower Riemann Sums
The lower Riemann sum, L(f, P ), and the upper Riemann sum, U (f, P ), for the function
f and the partition P are de
ned by:
n
X
L(f, P ) = f (l1 )∆x1 + · · · + f (ln )∆xn = f (li )∆xi ,
i=1
n
X
U (f, P ) = f (u1 )∆x1 + · · · + f (un )∆xn = f (ui )∆xi .
i=1
De
nition 3 The De
nite Integral
Suppose there is exactly one number I such that for every partition P of [a, b] we have
L(f, P ) ≤ I ≤ U (f, P ).
Then we say that the function f is integrable on [a, b], and we call I the de
nite integral
of f on [a, b]. The de
nite integral is denoted by the symbol
Z b
I= f (x)dx.
a
The dummy variable of the de
nite integral is x.
For all partitions P of [a, b], we have
Z b
L(f, P ) ≤ f (x)dx ≤ U (f, P )
a
Given a partition P having kP k = max ∆xi , chose a point ci (called a tag ) in each subinterval
and let c = (c1 , c2 , ..., cn ) denote the set of these tags. The sum
n
X
R(f, P, c) = f (ci )∆xi = f (c1 )∆x1 + · · · + f (cn )∆xn
i=1
2
, is called the Riemann sum of f on [a, b] corresponding to partition P and tags c.
The limit of a Riemann sum is the de
nite integral, that is
Z b
lim R(f, P, c) = f (x)dx
n(P )→∞, kP k→0 a
Theorem 5.2 If f is continuous on [a, b], then f is integrable on [a, b].
It is su
cient that, for any given , we should be able to
nd a partition P of [a, b] for which
U (f, P ) − L(f, P ) < , this restricts there to be only one I .
5.4 Properties of the De
nite Integral
If a > b, we have ∆xi < 0 for each i, so the integral will be negative for positive functions f and
vise versa.
Theorem 5.3 Properties of the De
nite Integral
Let f and g be integrable on an interval containing the points a, b, and c. Then
(a) An integral over an interval of zero length is zero.
Z a
f (x)dx = 0.
a
(b) Reversing the limits of integration changes the sign of the integral.
Z a Z b
f (x)dx = − f (x)dx.
b a
(c) An integral depends linearly on the integrand. If A and B are constants, then
Z b Z b Z b
(Af (x) + Bg(x))dx = A f (x)dx + B g(x)dx.
a a a
(d) An integral depends additively on the interval of integration.
Z b Z c Z c
f (x)dx + f (x)dx = f (x)dx.
a b a
(e) If a ≤ b and f (x) ≤ g(x) for a ≤ x ≤ b, then
Z b Z b
f (x)dx ≤ g(x)dx.
a a
(f) The triangle inequality for sums extends to de
nite integrals. If a ≤ b, then
Z b Z b
f (x)dx ≤ |f (x)|dx.
a a
(g) The integral of an odd function over an interval symmetric about zero is zero. If f is an
odd function, then Z a
f (x)dx = 0.
−a
3
