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Math 554 – Integration
Handout #9 – 4/12/
Defn. A collection of n + 1 distinct points of the interval [a, b]
P := {x 0 = a < x 1 < · · · < xi− 1 < xi < · · · < b =: xn}
is called a partition of the interval. In this case, we define the norm of the partition by ‖P ‖ := max 1 ≤i≤n ∆xi.
where ∆xi := xi − xi− 1 is the length of the i-th subinterval [xi− 1 , xi].
Defn. For a given partition P , we define the Riemann upper sum of a function f by
U (P, f ) :=
∑n i=
Mi ∆xi
where Mi denotes the supremum of f over each of the subintervals [xi− 1 , xi]. Sim- ilarly, we define the Riemann lower sum of a function f by
L(P, f ) :=
∑n i=
mi ∆xi
where mi denotes the infimum of f over each of the subintervals [xi− 1 , xi]. Since mi ≤ Mi, we note that L(P, f ) ≤ U (P, f ).
for any partition P.
Defn. Suppose P 1 , P 2 are both partitions of [a, b], then P 2 is called a refinement of P 1 , denoted by P 1 ≺ P 2 ,
if as sets P 1 ⊆ P 2.
Note. If P 1 ≺ P 2 , it follows that ‖P 2 ‖ ≤ ‖P 1 ‖ since each of the subintervals formed by P 2 is contained in a subinterval which arises from P 1.
Lemma. If P 1 ≺ P 2 , then L(P 1 , f ) ≤ L(P 2 , f ).
and U (P 2 , f ) ≤ U (P 1 , f ).
Proof. Suppose first that P 1 is a partition of [a, b] and that P 2 is the partition obtained from P 1 by adding an additional point z. The general case follows by induction, adding one point at at time. In particular, we let
P 1 := {x 0 = a < x 1 < · · · < xi− 1 < xi < · · · < b =: xn}
and P 2 := {x 0 = a < x 1 < · · · < xi− 1 < z < xi < · · · < b =: xn}
for some fixed i. We focus on the upper Riemann sum for these two partitions, noting that the inequality for the lower sums follows similarly. Observe that
U (P 1 , f ) :=
∑n j=
Mj ∆xj
and
U (P 2 , f ) :=
i∑− 1 j=
Mj ∆xj + M (z − xi− 1 ) + M˜ (xi − z) +
∑n j=i+
Mj ∆xj
where M := sup[xi− 1 ,z] f (x) and M˜ := sup[z,xi] f (x). It then follows that U (P 2 , f ) ≤ U (P 1 , f ) since M, M˜ ≤ Mi. 2
Defn. If P 1 and P 2 are arbitrary partitions of [a, b], then the common refinement of P 1 and P 2 is the formal union of the two.
Corollary. Suppose P 1 and P 2 are arbitrary partitions of [a, b], then
L(P 1 , f ) ≤ U (P 2 , f ).
Proof. Let P be the common refinement of P 1 and P 2 , then
L(P 1 , f ) ≤ L(P, f ) ≤ U (P, f ) ≤ U (P 2 , f ). 2
Defn. The lower Riemann integral of f over [a, b] is defined to be ∫ (^) b a
f (x)dx := sup all partitions P of [a,b]
L(P, f ).
Similarly, the upper Riemann integral of f over [a, b] is defined to be
∫ (^) b a f^ (x)dx^ :=^ all partitionsinf P of [a,b]
U (P, f ).
where the ξj , satisfying xj− 1 ≤ ξj ≤ xj (1 ≤ j ≤ n), are arbitrary.
Corollary. Suppose that f is Riemann integrable on [a, b], then there is a unique number γ ( =
∫ (^) b a f^ (x)dx) such that for every^ >^ 0 there exists a partition^ P^ of [a, b] such that if P ≺ P 1 , P 2 , then
i.) 0 ≤ U (P 1 , f ) − γ <
ii.) 0 ≤ γ − L(P 2 , f ) <
iii.) |γ − R(P 1 , f, ξ)| <
where R(P 1 , f, ξ) is any Riemann sum of f for the partition P 1. In this sense, we can interpret (^) ∫ b a f^ (x)dx^ =^ ‖Plim ‖→ 0 R(P, f,^ ξ).
although we would actually need to show a little more to be entirely correct. Proof. Since L(P 2 , f ) ≤ γ ≤ U (P 1 , f ) for all partitions, we see that parts i.) and ii.) follow from the definition of the Riemann integral. To see part iii.), we observe that mj ≤ f (ξj ) ≤ Mj and hence that
L(P 1 , f ) ≤ R(P 1 , f, ξ) ≤ U (P 1 , f ).
But we also know that both
L(P 1 , f ) ≤ γ ≤ U (P 1 , f )
and condition (*) hold, from which part iii.) follows. 2
Theorem. If f is continuous on [a, b], then f is Riemann-integrable on [a, b]. Proof. We use the condition (*) to prove that f is Riemann-integrable. If > 0, we set 0 := /(b − a). Since f is continuous on [a, b], f is uniformly continuous. Hence there is a δ > 0 such that |f (y) − f (x)| < 0 if |y − x| < δ. Suppose that ‖P ‖ < δ, then it follows that |Mi − mi| ≤ 0 (1 ≤ i ≤ n). Hence
U (P, f ) − L(P, f ) =
∑n i=
(Mi − mi)∆xi ≤ 0 (b − a) = . 2
Theorem. If f is monotone on [a, b], then f is Riemann-integrable on [a, b]. Proof. If f is constant, then we are done. We prove the case for f monotone increas- ing. The case for monotone decreasing is similiar. We again use the condition (*) to prove that f is Riemann-integrable. If > 0, we set δ := /(f (b) − f (a)) and consider any partition P with ‖P ‖ < δ. Since f is monotone increasing on [a, b], then Mi = f (xi) and mi = f (xi− 1 ). Hence
U (P, f ) − L(P, f ) =
∑n i=
(Mi − mi)∆xi
=
∑n i=
(f (xi) − f (xi− 1 ))∆xi
≤ ‖P ‖
∑n i=
(f (xi) − f (xi− 1 )) < δ (f (b) − f (a)) = . 2
Theorem. (Properties of the Riemann Integral) Suppose that f and g are Riemann integrable and k is a real number, then
i.) ∫ (^) b a k f^ (x)^ dx^ =^ k^
∫ (^) b a f^ (x)^ dx ii.) ∫ (^) b a f^ +^ g dx^ =^
∫ (^) b a f dx^ +^
∫ (^) b a g dx iii.) g ≤ f implies ∫ (^) b a g dx^ ≤^
∫ (^) b a f dx. iv.) | ∫ (^) b a f dx| ≤^
∫ (^) b a |f^ |^ dx
Proof. To prove part i.), we observe that in case k ≥ 0, then sup[xi− 1 ,xi] kf (x) = kMi and inf[xi− 1 ,xi] kf (x) = kmi. Hence U (P, kf ) = kU (P, f ) and L(P, kf ) = kL(P, f ). In the case that k < 0, then sup[xi− 1 ,xi] kf (x) = kmi and inf[xi− 1 ,xi] kf (x) = kMi. It follows in this case that U (P, kf ) = kL(P, f ) and L(P, kf ) = kU (P, f ) and so
∫ (^) b a k f^ (x)dx^ =^ k
∫ (^) b a
f (x)dx
which contains the points c and d. Let P ∗^ be the partition obtained by restricting the partition P˜ to the interval [c, d], then
U (P ∗, f ) − L(P ∗, f ) ≤ U ( P , f˜ ) − L( P , f˜ ) ≤ U (P, f ) − L(P, f ) <
and so f is Riemann integrable over [c, d]. To prove the identity (3), we use the fact that condition () holds when f is Riemann integrable. Let > 0, then for / 3 > 0, we may apply () to each of the intervals I = [a, b], [a, c] and [c, b], respectively, to obtain partitions PI which satisfy
(4) 0 ≤ UI (PI , f ) −
∫ I f dx^ ≤^ UI^ (PI^ , f^ )^ −^ LI^ (PI^ , f^ )^ < /^3.
We let P be the partition of [a, b] formed by the union of the two partitions P[a,c], P[c,b], and P˜ be the common refinement of P and P[a,b]. Observing that
(5) U[a,b]( P , f˜ ) = U[a,c]( P˜ 1 , f ) + U[c,b]( P˜ 2 , f ),
we can combine with inequality (4) to obtain ∣∣ ∣∫^ ac f dx + ∫^ cb f dx − ∫^ ab f dx
∣∣ ∣ (^) ≤
∣∣ ∣U[a,c]( P , f˜ ) − ∫^ ac f dx
∣∣ ∣ (^) +
∣∣ ∣U[c,b]( P , f˜ ) − ∫^ cb f dx
∣∣ ∣
∣∣ ∣U[a,b]( P , f˜ ) − ∫ (^) b a f dx
∣∣ ∣ < 3 0 = .
Since > 0 was arbitrary, then equality (3) must hold. 2
Theorem. (Intermediate Value Theorem for Integrals) If f is continuous on [a, b], then there exists ξ between a and b such that ∫ (^) b a f^ (x)^ dx^ =^ f^ (ξ)(b^ −^ a).
Proof. Since f is continuous on [a, b] and for η :=
∫ (^) b a f dx b − a
there holds
min [a,b] f (x) ≤ η ≤ max [a,b] f (x),
then by the Intermediate Value Theorem for continuous functions, there exists a ξ ∈ [a, b] such that f (ξ) = η. 2
Theorem. (Fundamental Theorem of Calculus, I. Derivative of an Integral) Sup- pose that f is continuous on [a, b] and set F (x) := ∫ (^) x a f^ (y)dy, then^ F^ is differentiable and F ′(x) = f (x) for a < x < b.
Proof. Notice that
F (x 0 + h) − F (x 0 ) h
∫ (^) x 0 +h x 0 f dx h
= f (ξ)
for some ξ between x 0 and x 0 + h. Hence, as h → 0, then ξ = ξh converges to x 0 and so the displayed difference quotient has a limit of f (x 0 ) as h → 0. 2
Theorem. (Fundamental Theorem of Calculus, Part II. Integral of a Derivative) Suppose that F is function with a continuous derivative on [a, b], then
∫ (^) b a F^
′(y) dy = F (x)|x=b x=a :=^ F^ (b)^ −^ F^ (a)
. Proof. Define G(x) :=
∫ (^) x a F^ ′(y) dy, and set H := F − G. Since the derivative of H is
identically zero by Part I of the Fundamental Theorem of Calculus, then the Mean Value Theorem implies that H(b) = H(a). Expressing this in terms of F and G gives
F (b) −
∫ (^) b a F^
′(y) dy = F (a),
which establishes the theorem. 2
