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. . 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. 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. 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