Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
5.2 Riemann Sums
Definition
Let f : [a, b] →R.
(i)
A Riemann sum of f with respect to a partition P = {x0, . . . ,xn} of [a,b] is a sum of the form
n
X
j=1
f (tj)(xj− xj−1),
where the choice of tj ∈ [xj−1,xj]is arbitrary.
5.2 Riemann Sums
Definition
Let f : [a, b] →R.
(i)
A Riemann sum of f with respect to a partition P = {x0, . . . ,xn} of [a,b] is a sum of the form
n
X
j=1
f (tj)(xj− xj−1),
where the choice of tj ∈ [xj−1,xj]is arbitrary.
5.2 Riemann Sums
Definition
Let f : [a, b] →R.
(i)
A Riemann sum of f with respect to a partition P = {x0, . . . ,xn} of [a,b] is a sum of the form
n
X
j=1
f (tj)(xj− xj−1),
where the choice of tj ∈ [xj−1,xj]is arbitrary.
5.2 Riemann Sums
Definition
Let f : [a, b] →R.
(i)
A Riemann sum of f with respect to a partition P = {x0, . . . ,xn} of [a,b] is a sum of the form
n
X
j=1
f (tj)(xj− xj−1),
where the choice of tj ∈ [xj−1,xj]is arbitrary.
Definition (ii)
The Riemann sums of f are said to converge to I(f) as
||P|| → 0 if and only if given > 0 there is a partition P of [a,b] such that
P = {x0, . . . ,xn} ⊇ P implies
n
X
j=1
f (tj)(xj− xj−1) −I(f )
<
for all choices of tj ∈ [xj−1,xj], j = 1, 2, . . . , n. In this case we shall use the notation
I(f ) = lim
||P||→0 n
X
j=1
f (tj)(xj− xj−1).
Definition (ii)
The Riemann sums of f are said to converge to I(f) as
||P|| → 0 if and only if given > 0 there is a partition P of [a,b] such that
P = {x0, . . . ,xn} ⊇ P implies
n
X
j=1
f (tj)(xj− xj−1) −I(f )
<
for all choices of tj ∈ [xj−1,xj], j = 1, 2, . . . , n. In this case we shall use the notation
I(f ) = lim
||P||→0 n
X
j=1
f (tj)(xj− xj−1).
Definition (ii)
The Riemann sums of f are said to converge to I(f) as
||P|| → 0 if and only if given > 0 there is a partition P of [a,b] such that
P = {x0, . . . ,xn} ⊇ P implies
n
X
j=1
f (tj)(xj− xj−1) −I(f )
<
for all choices of tj ∈ [xj−1,xj], j = 1, 2, . . . , n. In this case we shall use the notation
I(f ) = lim
||P||→0 n
X
j=1
f (tj)(xj− xj−1).
Theorem
Let a, b ∈R with a < b, and suppose that f : [a, b] → R is bounded. Then f is Riemann integrable on [a,b] if and only if
I(f ) = lim
||P||→0 =
n
X
j=1
f (tj)(xj− xj=1)
exists, in which case
I(f ) = Z b
a
f (x )dx
Theorem
Let a, b ∈R with a < b, and suppose that f : [a, b] → R is bounded. Then f is Riemann integrable on [a,b] if and only if
I(f ) = lim
||P||→0 =
n
X
j=1
f (tj)(xj− xj=1)
exists, in which case
I(f ) = Z b
a
f (x )dx
Theorem (Linear Property)
If f,g are integrable on [a,b] and α ∈R, then f + g and αf are integrable on [a,b]. In fact,
Z b a
(f (x ) + g(x ))dx = Z b
a
f (x )dx + Z b
a
g(x )dx and
Z b a
(αf (x ))dx = α Z b
a
f (x )dx .
Theorem (Linear Property)
If f,g are integrable on [a,b] and α ∈R, then f + g and αf are integrable on [a,b]. In fact,
Z b a
(f (x ) + g(x ))dx = Z b
a
f (x )dx + Z b
a
g(x )dx and
Z b a
(αf (x ))dx = α Z b
a
f (x )dx .
Theorem
If f is integrable on [a,b], then f is integrable on each subinterval [c,d] of [a,b]. Moreover,
Z b a
f (x )dx = Z c
a
f (x )dx + Z b
c
f (x )dx
for all c ∈ (a, b)
Theorem
If f is integrable on [a,b], then f is integrable on each subinterval [c,d] of [a,b]. Moreover,
Z b a
f (x )dx = Z c
a
f (x )dx + Z b
c
f (x )dx
for all c ∈ (a, b)
Theorem
If f is (Riemann) integrable on [a,b], then |f | is integrable on [a,b] and
Z b a
f (x )dx
≤ Z b
a
|f (x)|dx.
Theorem
If f is (Riemann) integrable on [a,b], then |f | is integrable on [a,b] and
Z b a
f (x )dx
≤ Z b
a
|f (x)|dx.
Corollary:
If f and g are (Riemann) integrable on [a,b], then so is fg.
Corollary:
If f and g are (Riemann) integrable on [a,b], then so is fg.
Theorem (First Mean Value Theorem For Integrals)
Suppose that f and g are integrable on [a,b] with g(x ) ≥ 0 for all x ∈ [a, b]. If
m = inf
x ∈[a,b]f (x ) and M = sup
x ∈[a,b]
f (x ),
then there is a number c ∈ [m, M] such that Z b
a
f (x )g(x )dx = c Z b
a
g(x )dx .
In particular, if f is continuous on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = f (x0) Z b
a
g(x )dx .
Theorem (First Mean Value Theorem For Integrals)
Suppose that f and g are integrable on [a,b] with g(x ) ≥ 0 for all x ∈ [a, b]. If
m = inf
x ∈[a,b]f (x ) and M = sup
x ∈[a,b]
f (x ),
then there is a number c ∈ [m, M] such that Z b
a
f (x )g(x )dx = c Z b
a
g(x )dx .
In particular, if f is continuous on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = f (x0) Z b
a
g(x )dx .
Theorem (First Mean Value Theorem For Integrals)
Suppose that f and g are integrable on [a,b] with g(x ) ≥ 0 for all x ∈ [a, b]. If
m = inf
x ∈[a,b]f (x ) and M = sup
x ∈[a,b]
f (x ),
then there is a number c ∈ [m, M] such that Z b
a
f (x )g(x )dx = c Z b
a
g(x )dx .
In particular, if f is continuous on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = f (x0) Z b
a
g(x )dx .
Theorem
If f is (Riemann) integrable on [a,b], then F (x ) = Z x
a
f (t)dt exists and is continuous on [a,b].
Theorem
If f is (Riemann) integrable on [a,b], then F (x ) = Z x
a
f (t)dt exists and is continuous on [a,b].
Theorem (Second Mean Value Theorem For Integrals) Suppose that f,g are integrable on [a,b], that g is
nonnegative on [a,b], and that m,M are real numbers that satisfy m ≤ inf f ([a, b]) and M ≥ sup f ([a, b]). Then there is an x0 ∈ [a, b] such that
Z b a
f (x )g(x )dx = m Z x0
a
g(x )dx + M Z b
x0
g(x )dx .
In particular, if f is also nonnegative on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = M Z b
x0
g(x )dx .
Theorem (Second Mean Value Theorem For Integrals) Suppose that f,g are integrable on [a,b], that g is
nonnegative on [a,b], and that m,M are real numbers that satisfy m ≤ inf f ([a, b]) and M ≥ sup f ([a, b]). Then there is an x0 ∈ [a, b] such that
Z b a
f (x )g(x )dx = m Z x0
a
g(x )dx + M Z b
x0
g(x )dx .
In particular, if f is also nonnegative on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = M Z b
x0
g(x )dx .
Theorem (Second Mean Value Theorem For Integrals) Suppose that f,g are integrable on [a,b], that g is
nonnegative on [a,b], and that m,M are real numbers that satisfy m ≤ inf f ([a, b]) and M ≥ sup f ([a, b]). Then there is an x0 ∈ [a, b] such that
Z b a
f (x )g(x )dx = m Z x0
a
g(x )dx + M Z b
x0
g(x )dx .
In particular, if f is also nonnegative on [a,b], then there is an x0 ∈ [a, b] that satisfies
Z b a
f (x )g(x )dx = M Z b
x0
g(x )dx .