Advanced Calculus (II)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
2009
Ch12: Integration on R
n12.2: 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|,
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
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.
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.
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.
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 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.
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.
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 .
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 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
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.
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.
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.
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
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.
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 ).
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.
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 ).
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.
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.
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.
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.
