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WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

2009

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## Ch12: Integration on R

n

### 12.2: Riemann Integration on Jordan Regions

Definition (12.15)

Let E be a Jordan region inRn, 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 toRnby setting f (x) = 0 for x ∈ Rn\E.

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

Mj|Rj|,

(3)

Definition (12.15)

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

Rj∩E6=∅

mj|Rj|,

where mj =infx∈Rjf (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

(4)

Remark (12.16)

Let E be a nonempty Jordan region inRn, 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.

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Remark (12.16)

Let E be a nonempty Jordan region inRn, 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.

(6)

Remark (12.16)

Let E be a nonempty Jordan region inRn, 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.

(7)

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.

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Remark (12.18)

Let E be a Jordan region inRn 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.

(9)

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 := {R1, . . . ,Rp} is any grid finer than G0and Mj,mj are defined as in Definition 12.15, then

(U) Z

E

f (x)d x − X

Rj⊂Eo

Mj|Rj|

< ε

and

(L) Z

E

f (x)d x − X

Rj⊂Eo

mj|Rj|

< ε.

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Theorem (12.21)

If E is a closed Jordan region inRn and f : E →R is continuous on E , then f is integrable on E .

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Theorem (12.22)

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

Z

E

1dx.

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Theorem (12.23 Linear Properties)

Let E be a Jordan region inRn, 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

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Theorem (12.23 Linear Properties)

Let E be a Jordan region inRn, 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.

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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∪ E2and

(12) Z

E1∪E2

f (x)d x = Z

E1

f (x)d x + Z

E2

f (x)d x.

(15)

Theorem (12.24)

Let E be a Jordan region inRn, 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.

(16)

Theorem (12.24)

Let E be a Jordan region inRn, 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

(17)

Theorem (12.24)

Let E be a Jordan region inRn, 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.

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Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region inRn, 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 ).

(19)

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region inRn, 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.

(20)

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region inRn, 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 ).

(21)

Theorem (12.25 Comparison Theorem for Multiple Integrals)

Let E be a Jordan region inRn, 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.

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Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region inRn 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∈Ef (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.

(23)

Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region inRn 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∈Ef (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.

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Theorem (12.26 Mean Value Theorem for Multiple Integrals)

Let E be a Jordan region inRn 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∈Ef (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.

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## 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