Arithmetic, geometric, harmonic, and quadratic means of functions... The arithmetic mean, geometric mean, harmonic mean, and quadratic mean of a functionf x on an interval [a, b] are int
Trang 17 H¨older’s inequality (at p =2, it translates into Bunyakovsky’s inequality):
a b f (x)g(x) dx
≤
b
a |f(x)|p dx1
p b
a |g(x)|p– p1 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) and g(x) are monotonically increasing functions and h(x) is a positive integrable function on [a, b].
9 Jensen’s inequality:
f
7b
a7g b (t)x(t) dt
a g (t) dt
≤
7b
a g7(t)f (x(t)) dt b
a g (t) dt
if f (x) is convex (f >0);
f
7b
a7g b (t)x(t) dt
a g (t) dt
≥
7b
a g7(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 ) and g(t)≥ 0 The equality is attained if
and only if either x(t) = const or f (x) is a linear function Jensen’s inequality serves as a
general source for deriving various integral inequalities
10 Steklov’s inequality Let f (x) be a continuous function on [0, π] and let it have
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)]2dx≥ π
0 [f (x)]
2dx.
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 If a >0and f (x)≥ 0on [0, a], then
a
0 f (x) dx
4
≤π2 a
0 f
2(x) dx a
0 x
2f2(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) on a discrete
set of points:
f kn = f (a + kδ n), δ n= b – a n (k =1, , n).
Trang 2The arithmetic mean, geometric mean, harmonic mean, and quadratic mean of a function
f (x) on an interval [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
f2
kn
1 2
=
1
b – a
b
a f
2(x) dx1 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 factor1/√ b – a.
The following inequalities hold:
(b – a)
b
a
dx
f (x)
– 1
≤ exp
1
b – a
b
a ln f (x) dx
b – a
b
a
f (x) dx≤
1
b – a
b
a
f2(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 range0 ≤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) is the inverse of g(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) = ln y,
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) = y2
Trang 37.2.6 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) and y = g(x) and straight lines
x = a and x = b in the x, y plane (see Fig 7.2 a) 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
y=f x( )
ρ=f( )φ
α β
y=g x( )
y
x
O
Figure 7.2 (a) A domain D bounded by two curves y = f (x) and y = g(x) on an interval [a, b]; (b) a curvilinear
sector.
2 Area of a domain D Let x = x(t) and y = y(t), with t1≤t≤t2, be parametric equations
of a piecewise-smooth simple closed curve bounding on its left (traced counterclockwise)
a domain D with area S Then
S = –
t2
t1
y (t)x (t) dt =
t2
t1
x (t)y (t) dt = 1
2
t2
t1
x (t)y (t) – y(t)x (t)
dt
3 Area of a curvilinear sector Let a curve ρ = 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)]
2dϕ.
4 Area of a surface of revolution Let a surface of revolution be generated by rotating
a curve y = 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)]2dx
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)]
2dx.
Trang 4y=f x( )
y
x
z
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)]2dx
(b) If a plane curve is defined parametrically by equations x = x(t) and y = y(t), with
t[α, β] and x(t) and y(t) being continuously differentiable functions, then its length is
calculated by
L=
β
α
[x (t)]2+ [y (t)]2dt
(c) If a curve is defined 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
ρ2(ϕ) + [ρ (ϕ)]2dϕ.
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] and x(t), y(t), and z(t) being continuously differentiable
functions, is calculated by
L=
β
α
[x (t)]2+ [y (t)]2+ [z (t)]2dt
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.
Trang 52 Mass of a rectilinear rod of variable density Suppose a rod with a constant cross-sectional area S occupies an interval [0, l] on the x-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)]2dx
If the shape of the rod is defined parametrically by x = x(t) and y = y(t), then its mass
is found as
m = S
b
a ρ x (t), y(t)
[x (t)]2+ [y (t)]2dt
4 The coordinates of the center of mass of a plane homogeneous material curve whose shape is defined by an equation y = f (x), with a≤x≤b, are calculated by the formulas
xc= 1
L
b
a x
1+ [y (x)]2dx, yc= 1
L
b
a f (x)
1+ [y (x)]2dx,
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
xc= 1
L
b
a x (t)
[x (t)]2+ [y (t)]2dt, yc = 1
L
b
a y (t)
[x (t)]2+ [y (t)]2dt
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◦ Let y = f (x) be a function defined 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
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.
Trang 62◦ 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, we have
lim
x→∞ F (x) = 1
1– λ x→∞lim 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) = a1–λ
λ– 1, and if λ≤ 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
THEOREM1 (CAUCHY’S CONVERGENCE CRITERION) For the integral (7.2.7.1) to be
convergent it is necessary and sufficient that for any ε >0there exists a number R such that
the inequality
α β f (x) dx
< ε
holds for any β > α > R.
THEOREM2 If 0 ≤ f (x) ≤ g (x) for x ≥ a, then the convergence of the integral
∞
a g (x) dximplies the convergence of the integral ∞
a f (x) dx; moreover, ∞
a f (x) dx≤
∞
a g (x) dx If the integral ∞
a f (x) dxis divergent, then the integral ∞
a g (x) dxis also divergent
Trang 7THEOREM3 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
x2 dxis absolutely convergent, since sin x
x2
≤ 1
x2 and the integral
1
1
x2 dxis convergent (see Example 1).
THEOREM4 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,
∞
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
THEOREM5 (COROLLARY OFTHEOREM4) 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) dxis convergent; (ii) if
λ≤ 1and ϕ(x)≥c>0, then the integral is divergent
THEOREM6 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, ∞).
THEOREM7 (ANALOGUE OFABEL’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 ∞
is convergent
THEOREM8 (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
... 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... application of the integral1 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... 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