## 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 ∈ [x_{j−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 ∈ [x_{j−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 ∈ [x_{j−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 ∈ [x_{j−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− x_{j−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− x_{j−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− x_{j−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− x_{j=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− x_{j=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 .