## Advanced Calculus (II)

W^{EN}-C^{HING}L^{IEN}

Department of Mathematics National Cheng Kung University

2009

## Ch12: Integration on **R**

^{n}

### 12.2: Riemann Integration on Jordan Regions

Definition (12.15)

Let E be a Jordan region in**R**^{n}, let f : E →**R be a**
bounded function, let R be an n-dimensional rectangle
such that E ⊆ R, and let G = {R1, . . . ,Rp} be a grid on R.

Extend f to**R**^{n}by setting f (x) = 0 for x ∈ R^{n}\E.

(i) The upper sum of f on E with respect to G is U(f , G) := X

Mj|R_{j}|,

Definition (12.15)

(ii) The lower sum of f on E with respect to G is L(f , G) := X

R_{j}∩E6=∅

mj|R_{j}|,

where mj =inf**x∈R**jf (x).

(iii) The upper and lower integrals of f on E are defined by (L)

Z

E

f (x)d x := (L) Z

E

fdV := sup

G

L(f , G)

and

(U) Z

E

f (x)d x := (U) Z

E

fdV := inf

G U(f , G),

where the supremum and infimum are taken over all grids

Remark (12.16)

Let E be a nonempty Jordan region in**R**^{n}, let f : E →**R**
be bounded, and let R be a rectangle that contains E .
(i) If G and H are grids on R, then L(f , G) ≤ U(f , H).

(ii) The upper and lower integrals of f over E exist, do not depend on the choice of R, and satisfy

(6) (L)

Z

E

f (x)d x ≤ (U) Z

E

f (x)d x.

Remark (12.16)

Let E be a nonempty Jordan region in**R**^{n}, let f : E →**R**
be bounded, and let R be a rectangle that contains E .
(i) If G and H are grids on R, then L(f , G) ≤ U(f , H).

(ii) The upper and lower integrals of f over E exist, do not depend on the choice of R, and satisfy

(6) (L)

Z

E

f (x)d x ≤ (U) Z

E

f (x)d x.

Remark (12.16)

Let E be a nonempty Jordan region in**R**^{n}, let f : E →**R**
be bounded, and let R be a rectangle that contains E .
(i) If G and H are grids on R, then L(f , G) ≤ U(f , H).

(ii) The upper and lower integrals of f over E exist, do not depend on the choice of R, and satisfy

(6) (L)

Z

E

f (x)d x ≤ (U) Z

E

f (x)d x.

Definition (12.17)

A real-valued bounded function f defined on a Jordan region E is said to be (Riemann) integrable on E if and only if for every ε > 0 there is a grid G such that

U(f , G) − L(f , G) < ε.

By modifying the proof of Theorem 5.15, we can establish the following result.

Remark (12.18)

Let E be a Jordan region in**R**^{n} and suppose that
f : E →**R is bounded. Then f is integrable on E if and**
only if

(7) (L)

Z

E

f (x)d x = (U) Z

E

f (x)d x.

Theorem (12.20)

Let E be a Jordan region and suppose that f : E →**R is**
bounded. Then given ε > 0 there is a grid G0such that if
G := {R_{1}, . . . ,Rp} is any grid finer than G_{0}and Mj,mj are
defined as in Definition 12.15, then

(U) Z

E

f (x)d x − X

Rj⊂E^{o}

Mj|R_{j}|

< ε

and

(L) Z

E

f (x)d x − X

Rj⊂E^{o}

mj|R_{j}|

< ε.

Theorem (12.21)

If E is a closed Jordan region in**R**^{n} and f : E →**R is**
continuous on E , then f is integrable on E .

Theorem (12.22)

If E is a closed Jordan region, then Vol(E ) =

Z

E

1dx.

Theorem (12.23 Linear Properties)

Let E be a Jordan region in**R**^{n}, let f , g : E →**R, and let α**
be a scalar.

(i) If f , g are integrable on E , then so are αf and f + g. In fact,

(10)

Z

E

αf (x)d x = α Z

E

f (x)d x and

Z Z Z

Theorem (12.23 Linear Properties)

Let E be a Jordan region in**R**^{n}, let f , g : E →**R, and let α**
be a scalar.

(i) If f , g are integrable on E , then so are αf and f + g. In fact,

(10)

Z

E

αf (x)d x = α Z

E

f (x)d x

and (11)

Z

E

(f (x)+g(x))d x = Z

E

f (x)d x+

Z

E

g(x)d x.

Theorem (12.23 Linear Properties)

(ii) If E1,E2⊆ E are nonoverlapping Jordan regions and f
is integrable on both E1 and E2, then f is integrable on
E1∪ E_{2}and

(12) Z

E1∪E2

f (x)d x = Z

E1

f (x)d x + Z

E2

f (x)d x.

Theorem (12.24)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are bounded functions.**

(i) If E0 is of volume zero, then g is integrable on E0 and Z

E0

g(x)d x = 0.

(ii) If f is integrable on E and if there is a subset E0of E such that Vol(E0) = 0 and f (x) = g(x) for all x ∈ E \E0, then g is integrable on E and

Z

E

g(x)d x = Z

E

f (x)d x.

Theorem (12.24)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are bounded functions.**

(i) If E0 is of volume zero, then g is integrable on E0 and Z

E0

g(x)d x = 0.

(ii) If f is integrable on E and if there is a subset E0of E such that Vol(E0) = 0 and f (x) = g(x) for all x ∈ E \E0, then g is integrable on E and

Theorem (12.24)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are bounded functions.**

(i) If E0 is of volume zero, then g is integrable on E0 and Z

E0

g(x)d x = 0.

(ii) If f is integrable on E and if there is a subset E0of E such that Vol(E0) = 0 and f (x) = g(x) for all x ∈ E \E0, then g is integrable on E and

Z

E

g(x)d x = Z

E

f (x)d x.

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are integrable on E .**

(i) If f (x) ≤ g(x) for E, then Z

E

f (x)d x ≤ Z

E

g(x)d x.

(ii) If m, M are scalars that satisfy m ≤ f (x) ≤ M for x ∈ E , then

mVol(E ) ≤ Z

E

f (x)d x ≤ MVol(E ).

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are integrable on E .**

(i) If f (x) ≤ g(x) for E, then Z

E

f (x)d x ≤ Z

E

g(x)d x.

(ii) If m, M are scalars that satisfy m ≤ f (x) ≤ M for x ∈ E , then

mVol(E ) ≤ Z

E

f (x)d x ≤ MVol(E ).

(iii) The function |f | is integrable on E and (17)

Z

f (x)d x ≤

Z

**|f (x)|dx.**

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are integrable on E .**

(i) If f (x) ≤ g(x) for E, then Z

E

f (x)d x ≤ Z

E

g(x)d x.

(ii) If m, M are scalars that satisfy m ≤ f (x) ≤ M for x ∈ E , then

mVol(E ) ≤ Z

E

f (x)d x ≤ MVol(E ).

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n}, and suppose that f , g :
E →**R are integrable on E .**

(i) If f (x) ≤ g(x) for E, then Z

E

f (x)d x ≤ Z

E

g(x)d x.

(ii) If m, M are scalars that satisfy m ≤ f (x) ≤ M for x ∈ E , then

mVol(E ) ≤ Z

E

f (x)d x ≤ MVol(E ).

(iii) The function |f | is integrable on E and (17)

Z

f (x)d x ≤

Z

**|f (x)|dx.**

Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n} and let f , g : E →**R be**
integrable on E with g(x) ≥ 0 for all x ∈ E .

(i) There is a number c satisfying

(19) inf

**x∈E**f (x) ≤ c ≤ sup

**x∈E**

f (x)

such that

(20) c

Z

E

g(x)d x = Z

E

f (x)g(x)d x.

Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n} and let f , g : E →**R be**
integrable on E with g(x) ≥ 0 for all x ∈ E .

(i) There is a number c satisfying

(19) inf

**x∈E**f (x) ≤ c ≤ sup

**x∈E**

f (x)

such that

(20) c

Z

E

g(x)d x = Z

E

f (x)g(x)d x.

(ii) There is a number c satisfying (19) such that cVol(E ) =

Z

f (x)d x.

Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region in**R**^{n} and let f , g : E →**R be**
integrable on E with g(x) ≥ 0 for all x ∈ E .

(i) There is a number c satisfying

(19) inf

**x∈E**f (x) ≤ c ≤ sup

**x∈E**

f (x)

such that

(20) c

Z

E

g(x)d x = Z

E

f (x)g(x)d x.