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WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

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## 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.

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## 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.

(4)

## 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)

## 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.

(6)

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).

(7)

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).

(8)

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).

(9)

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

(10)

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

(11)

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 .

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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 .

(13)

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)

(14)

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)

(15)

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.

(16)

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.

(17)

### Corollary:

If f and g are (Riemann) integrable on [a,b], then so is fg.

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### Corollary:

If f and g are (Riemann) integrable on [a,b], then so is fg.

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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 .

(20)

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 .

(21)

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 .

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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].

(23)

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].

(24)

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 .

(25)

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 .

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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 .

(27)

## Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung