Handbook of mathematics for engineers and scienteists part 48 ppsx

7.2. DEFINITE INTEGRAL 297
7. H
¨
older’s inequality (at p = 2, it translates into Bunyakovsky’s inequality):





b
a
f(x)g(x) dx




≤


b
a
|f(x)|
p
dx

1
p


b
a

|g(x)|
p
p–1
dx

p–1
p
, p > 1.
8. Chebyshev’s inequality:


b
a
f(x)h(x) dx


b
a
g(x)h(x) dx

≤


b
a
h(x) dx


b
a

f(x)g(x)h(x) dx

,
where f(x)andg(x) are monotonically increasing functions and h(x) is a positive integrable
function on [a, b].
9. Jensen’s inequality:
f


b
a
g(t)x(t) dt

b
a
g(t) dt

≤

b
a
g(t)f(x(t)) dt

b
a
g(t) dt
if f(x)isconvex(f

> 0);
f



b
a
g(t)x(t) dt

b
a
g(t) dt

≥

b
a
g(t)f(x(t)) dt

b
a
g(t) dt
if f(x) is concave (f

< 0),
where x(t) is a continuous function (a ≤ x ≤ b)andg(t) ≥ 0. The equality is attained if
and only if either x(t)=constorf(x) is a linear function. Jensen’s inequality serves as a
general source for deriving various integral inequalities.
10. Steklov’s inequality.Letf(x) be a continuous function on [0, π]andletithave
everywhere on [0, π], except maybe at finitely many points, a square integrable deriva-
tive f

(x). If either of the conditions

(a) f(0)=f(π)=0,
(b)

π
0
f(x) dx = 0
is satisfied, then the following inequality holds:

π
0
[f

(x)]
2
dx ≥

π
0
[f(x)]
2
dx.
The equality is only attained for functions f(x)=A sin x in case (a) and functions f(x)=
B cos x in case (b).
11. A π-related inequality.Ifa > 0 and f (x) ≥ 0 on [0, a], then


a
0
f(x) dx


4
≤ π
2


a
0
f
2
(x) dx


a
0
x
2
f
2
(x) dx

.
7.2.5-3. Arithmetic, geometric, harmonic, and quadratic means of functions.
Let f (x) be a positive function integrable on [a, b]. Consider the values of f(x)onadiscrete
set of points:
f
kn
= f (a + kδ
n
), δ
n

=
b – a
n
(k = 1, , n).
298 INTEGRALS
The arithmetic mean, geometric mean, harmonic mean, and quadratic mean of a function
f(x)onaninterval[a, b] are introduced using the definitions of the respective mean values
for finitely many numbers (see Subsection 1.6.1) and going to the limit as n →∞.
1. Arithmetic mean of a function f(x) on [a, b]:
lim
n→∞
1
n
n

k=1
f
kn
=
1
b – a

b
a
f(x) dx.
This definition is in agreement with another definition of the mean value of a function f(x)
on [a, b] given in Theorem 1 from Paragraph 7.2.5-1.
2. Geometric mean of a function f(x) on [a, b]:
lim
n→∞


n

k=1
f
kn

1/n
=exp

1
b – a

b
a
ln f (x) dx

.
3. Harmonic mean of a function f(x) on [a, b]:
lim
n→∞
n

n

k=1
1
f
kn


–1
=(b – a)


b
a
dx
f(x)

–1
.
4. Quadratic mean of a function f (x) on [a, b]:
lim
n→∞

1
n
n

k=1
f
2
kn

1/2
=

1
b – a


b
a
f
2
(x) dx

1/2
.
This definition differs from the common definition of the norm of a square integrable
function given in Paragraph 7.2.13-2 by the constant factor 1/
√
b – a.
The following inequalities hold:
(b –a)


b
a
dx
f(x)

–1
≤ exp

1
b – a

b
a
ln f(x) dx


≤
1
b – a

b
a
f(x) dx ≤

1
b – a

b
a
f
2
(x) dx

1/2
.
To make it easier to remember, let us rewrite these inequalities in words as
harmonic mean ≤ geometric mean ≤ arithmetic mean ≤ quadratic mean .
The equality is attained for f(x) = const only.
7.2.5-4. General approach to defining mean values.
Let g(y) be a continuous monotonic function defined in the range 0 ≤ y < ∞.
The mean of a function f(x) with respect to a function g(x) on an interval [a, b]is
defined as
lim
n→∞
g

–1

1
n
n

k=1
g(f
kn
)

= g
–1

1
b – a

b
a
g

f(x)

dx

,
where g
–1
(z)istheinverseofg(y).
The means presented in Paragraph 7.2.5-3 are special cases of the mean with respect to

a function:
arithmetic mean of f(x) = mean of f(x) with respect to g(y)=y,
geometric mean of f(x) = mean of f(x) with respect to g(y)=lny,
harmonic mean of f(x) = mean of f(x) with respect to g(y)=1/y,
quadratic mean of f (x) = mean of f(x) with respect to g(y)=y
2
.
7.2. DEFINITE INTEGRAL 299
7.2.6. Geometric and Physical Applications of the Definite Integral
7.2.6-1. Geometric applications of the definite integral.
1. The area of a domain D bounded by curves y = f(x)andy = g(x) and straight lines
x = a and x = b in the x, y plane(seeFig.7.2a) is calculated by the formula
S =

b
a

f(x)–g(x)

dx.
If g(x) ≡ 0, this formula gives the area of a curvilinear trapezoid bounded by the x-axis, the
curve y = f (x), and the straight lines x = a and x = b.
D
yfx= ()
()a ()b
ρφf= ()
α
β
ygx= ()
y

x
ab
O
Figure 7.2. (a) A domain D bounded by two curves y = f(x)andy = g(x)onaninterval[a, b]; (b) a curvilinear
sector.
2. Area of a domain D.Letx = x(t)andy =y(t), with t
1
≤ t ≤ t
2
, be parametric equations
of a piecewise-smooth simple closed curve bounding on its left (traced counterclockwise)
a domain D with area S.Then
S =–

t
2
t
1
y(t)x

(t) dt =

t
2
t
1
x(t)y

(t) dt =
1

2

t
2
t
1

x(t)y

(t)–y(t)x

(t)

dt.
3. Area of a curvilinear sector.Letacurveρ = f(ϕ), with ϕ
[α, β], be defined in the
polar coordinates ρ, ϕ. Then the area of the curvilinear sector {α ≤ ϕ ≤ β; 0 ≤ ρ ≤ f(ϕ)}
(see Fig. 7.2 b) is calculated by the formula
S =
1
2

β
α
[f(ϕ)]
2
dϕ.
4. Area of a surface of revolution. Let a surface of revolution be generated by rotating
acurvey = f(x) ≥ 0, x
[a, b], about the x-axis; see Fig. 7.3. The area of this surface is

calculated as
S = 2π

b
a
f(x)

1 +[f

(x)]
2
dx.
5. Volume of a body of revolution. Let a body of revolution be obtained by rotating
about the x-axis a curvilinear trapezoid bounded by a curve y = f(x), the x-axis, and straight
lines x = a and x = b; see Fig. 7.3. Then the volume of this body is calculated as
V = π

b
a
[f(x)]
2
dx.
300 INTEGRALS
yfx= ()
y
x
z
ab
O
Figure 7.3. A surface of revolution generated by rotating a curve y = f(x).

6. Arc length of a plane curve defined in different ways.
(a) If a curve is the graph of a continuously differentiable function y = f(x), x [a, b],
then its length is determined as
L =

b
a

1 +[f

(x)]
2
dx.
(b) If a plane curve is defined parametrically by equations x = x(t)andy = y(t), with
t
[α, β]andx(t)andy(t) being continuously differentiable functions, then its length is
calculated by
L =

β
α

[x

(t)]
2
+[y

(t)]
2

dt.
(c)Ifacurveisdefined in the polar coordinates ρ, ϕ by an equation ρ = ρ(ϕ), with
ϕ
[α, β], then its length is found as
L =

β
α

ρ
2
(ϕ)+[ρ

(ϕ)]
2
dϕ.
7. The arc length of a spatial curve defined parametrically by equations x = x(t),
y =y(t), and z = z(t), with t
[α, β]andx(t), y(t), and z(t) being continuously differentiable
functions, is calculated by
L =

β
α

[x

(t)]
2
+[y


(t)]
2
+[z

(t)]
2
dt.
7.2.6-2. Physical application of the integral.
1. Work of a variable force. Suppose a point mass moves along the x-axis from a point
x = a to a point x = b under the action of a variable force F(x) directed along the x-axis.
The mechanical work of this force is equal to
A =

b
a
F (x) dx.
7.2. DEFINITE INTEGRAL 301
2. Mass of a rectilinear rod of variable density. Suppose a rod with a constant cross-
sectional area S occupies an interval [0, l]onthex-axis and the density of the rod material
is a function of x: ρ = ρ(x). The mass of this rod is calculated as
m = S

l
0
ρ(x) dx.
3. Mass of a curvilinear rod of variable density. Let the shape of a plane curvilinear rod
with a constant cross-sectional area S be defined by an equation y = f(x), with a ≤ x ≤ b,
and let the density of the material be coordinate dependent: ρ = ρ(x, y). The mass of this
rod is calculated as

m = S

b
a
ρ

x, f(x)


1 +[y

(x)]
2
dx.
If the shape of the rod is defined parametrically by x = x(t)andy = y(t), then its mass
is found as
m = S

b
a
ρ

x(t), y(t)


[x

(t)]
2
+[y


(t)]
2
dt.
4. The coordinates of the center of mass of a plane homogeneous material curve whose
shape is definedbyanequationy = f(x), with a ≤ x ≤ b, are calculated by the formulas
x
c
=
1
L

b
a
x

1 +[y

(x)]
2
dx, y
c
=
1
L

b
a
f(x)


1 +[y

(x)]
2
dx,
where L is the length of the curve.
If the shape of a plane homogeneous material curve is defined parametrically by x = x(t)
and y = y(t), then the coordinates of its center of mass are obtained as
x
c
=
1
L

b
a
x(t)

[x

(t)]
2
+[y

(t)]
2
dt, y
c
=
1

L

b
a
y(t)

[x

(t)]
2
+[y

(t)]
2
dt.
7.2.7. Improper Integrals with Infinite Integration Limit
An improper integral is an integral with an infinite limit (limits) of integration or an integral
of an unbounded function.
7.2.7-1. Integrals with infinite limits.
1
◦
.Lety = f(x)beafunctiondefined and continuous on an infinite interval a ≤ x < ∞.If
there exists a finite limit lim
b→∞

b
a
f(x) dx, then it is called a (convergent) improper integral
of f (x) on the interval [a, ∞) and is denoted


∞
a
f(x) dx. Thus, by definition

∞
a
f(x) dx = lim
b→∞

b
a
f(x) dx.(7.2.7.1)
If the limit is infinite or does not exist, the improper integral is called divergent.
The geometric meaning of an improper integral is that the integral

∞
a
f(x) dx, with
f(x) ≥ 0, is equal to the area of the unbounded domain between the curve y = f (x), its
asymptote y = 0, and the straight line x = a on the left.
302 INTEGRALS
2
◦
. Suppose an antiderivative F(x) of the integrand function f (x) is known. Then the
improper integral (7.2.7.1) is
(i) convergent if there exists a finite limit lim
x→∞
F (x)=F (∞);
(ii) divergent if the limit is infinite or does not exist.
In case (i), we have


∞
a
f(x) dx = F (x)


∞
a
= F (∞)–F (a).
Example 1. Let us investigate the issue of convergence of the improper integral I =

∞
a
dx
x
λ
, a > 0.
The integrand f(x)=x
–λ
has an antiderivative F (x)=
1
1 – λ
x
1–λ
. Depending on the value of the parameter
λ,wehave
lim
x→∞
F (x)=
1

1 – λ
lim
x→∞
x
1–λ
=

0 if λ > 1,
∞ if λ ≤ 1.
Therefore, if λ > 1, the integral is convergent and is equal to I = F (∞)–F (a)=
a
1–λ
λ – 1
,andifλ ≤ 1,the
integral is divergent.
3
◦
. Improper integrals for other infinite intervals are defined in a similar way:

b
–∞
f(x) dx = lim
a→–∞

b
a
f(x) dx,

∞
–∞

f(x) dx =

c
–∞
f(x) dx +

∞
c
f(x) dx.
Note that if either improper integral on the right-hand side of the latter relation is convergent,
then, by definition, the integral on the left-hand side is also convergent.
4
◦
. Properties 2–4 and 6–9 from Paragraph 7.2.2-2, where a can be equal to –∞ and b can
be ∞, apply to improper integrals as well; it is assumed that all quantities on the right-hand
sides exist (the integrals are convergent).
7.2.7-2. Sufficient conditions for convergence of improper integrals.
In many problems, it suffices to establish whether a given improper integral is convergent
or not and, if yes, evaluate it. The theorems presented below can be useful in doing so.
T
HEOREM 1(CAUCHY’S CONVERGENCE CRITERION).
For the integral (7.2.7.1) to be
convergent it is necessary and sufficient that for any
ε > 0
there exists a number
R
such that
the inequality






β
α
f(x) dx




< ε
holds for any
β > α > R
.
THEOREM 2.
If
0 ≤ f (x) ≤ g(x)
for
x ≥ a
, then the convergence of the integral

∞
a
g(x) dx
implies the convergence of the integral

∞
a
f(x) dx
; moreover,


∞
a
f(x) dx ≤

∞
a
g(x) dx
. If the integral

∞
a
f(x) dx
is divergent, then the integral

∞
a
g(x) dx
is also
divergent.
7.2. DEFINITE INTEGRAL 303
T
HEOREM 3.
If the integral

∞
a
|f(x)|dx
is convergent, then the integral


∞
a
f(x) dx
is also convergent; in this case, the latter integral is called absolutely convergent.
Example 2. The improper integral

∞
1
sin x
x
2
dx is absolutely convergent, since



sin x
x
2



≤
1
x
2
and the
integral

∞
1

1
x
2
dx is convergent (see Example 1).
THEOREM 4.
Let
f(x)
and
g(x)
be integrable functions on any finite interval
a ≤ x ≤ b
and let there exist a limit, finite or infinite,
lim
x→∞
f(x)
g(x)
= K.
Then the following assertions hold:
1. If
0 < K < ∞
, both integrals

∞
a
f(x) dx,

∞
a
g(x) dx (
7

.2.7.2)
are convergent or divergent simultaneously.
2. If
0 ≤ K < ∞
, the convergence of the latter integral in (7.2.7.2) implies the conver-
gence of the former integral.
3. If
0 < K ≤ ∞
, the divergence of the latter integral in (7.2.7.2) implies the divergence
of the former integral.
THEOREM 5(COROLLARY OF THEOREM 4).
Given a function
f(x)
, let its asymptotics
for sufficiently large
x
have the form
f(x)=
ϕ(x)
x
λ
(λ > 0).
Then: (i) if
λ > 1
and
ϕ(x) ≤ c < ∞
, then the integral

∞
a

f(x) dx
is convergent; (ii) if
λ ≤ 1
and
ϕ(x) ≥ c > 0
, then the integral is divergent.
THEOREM 6.
Let
f(x)
be an absolutely integrable function on an interval
[a, ∞)
and let
g(x)
be a bounded function on
[a, ∞)
. Then the product
f(x)g(x)
is an absolutely integrable
function on
[a, ∞)
.
THEOREM 7(ANALOGUE OF ABEL’S TEST FOR CONVERGENCE OF INFINITE SERIES).
Let
f(x)
be an integrable function on an interval
[a, ∞)
such that the integral (7.2.7.1) is
convergent (maybe not absolutely) and let
g(x)
be a monotonic and bounded function on

[a, ∞)
. Then the integral

∞
a
f(x)g(x) dx (
7
.2.7.3)
is convergent.
THEOREM 8(ANALOGUE OF DIRICHLET’S TEST FOR CONVERGENCE OF INFINITE SE-
RIES).
Let (i)
f(x)
be an integrable function on any finite interval
[a, A]
and





A
a
f(x) dx




≤ K < ∞ (a ≤ A < ∞);
(ii)

g(x)
be a function tending to zero monotonically as
x →∞
:
lim
x→∞
g(x)=0
. Then the
integral (7.2.7.3) is convergent.