Additional Properties of the Definite Integral I.The integral of a nonnegative continuous function is nonnegative 5.81) if f(x)0 for all x e[a,b]. fx)dr≥0. The integral of a positive continuous function is positive: 6 5.82) if f(x)>0 for all x e la,bl. then f)d>0. Main Meny
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Additional Properties of the Definite Integral I. The integral of a nonnegative continuous function is nonnegative: The integral of a positive continuous function is positive:
II.The integral is order-preserving:for continuous functions fand g, 5.83) if f(x)<g(x)for allx e [a,b].then fe)dk≤g)d 5.840 f间<e(r)forall.n∫d<[gth MnMe 50
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. II. The integral is order-preserving: for continuous functions f and g
Additional Properties of the Definite Integral III.Just as the absolute value of a sum of numbers is less than or equal to the sum of the absolute values of those numbers, x1+x2+··+xl≤xl+x2+··+xl, the absolute value of an integral of a continuous function is less than or equal to the integral of the absolute value of that function: (5.8.5) f(x)dxs If(x)dx Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Additional Properties of the Definite Integral III. Just as the absolute value of a sum of numbers is less than or equal to the sum of the absolute values of those numbers, |x1 + x2 +· · ·+ xn | ≤ |x1 | + |x2 |+· · ·+|xn |, the absolute value of an integral of a continuous function is less than or equal to the integral of the absolute value of that function:
IV.Iffis continuous on [a,b],then (5.8.6 m(b-a）≤ fx)dk≤Mb-a) where m is the minimum value of fon [a,b]and Mis the maximum. Reasoning:m(b-a)is a lower sum for fand M(b-a)is an upper sum. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. IV. If f is continuous on [a, b], then where m is the minimum value of f on [a, b] and M is the maximum. Reasoning: m(b − a) is a lower sum for f and M(b − a) is an upper sum
Additional Properties of the Definite Integral V.Iffis continuous on [a,b]and u is a differentiable function ofx with values in [a,bl,then for all u(x)(a,b) d (5.8.7) f(t)dt =f(u(x)u(x). Example 1 Find」 Main Meny c墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Additional Properties of the Definite Integral V. If f is continuous on [a, b] and u is a differentiable function of x with values in [a, b], then for all u(x) (a, b) Example 1 Find 3 0 1 1 d x dt dx t +
Additional Properties of the Definite Integral Example 2 Find Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Additional Properties of the Definite Integral Example 2 + . 1 1 Find 2 2 dt dx t d x x
Additional Properties VI.Now a few words about the role of symmetry in integration.Suppose thatf is continuous on an interval of the form [-a,a],a closed interval symmetric about the origin (5.8.8) (a)if f is odd on [-a,a].then f(x)dx=0. (b)if f is even on [-a.a].then f(x)dx=2 f(x)dx. Jo Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Additional Properties VI. Now a few words about the role of symmetry in integration. Suppose that f is continuous on an interval of the form [−a, a], a closed interval symmetric about the origin
21 a 21 02 92 -q fodd feven Figure 5.8.1 Figure 5.8.2 Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Mean-Value Theorems for Integrals THEOREM 5.9.1 THE FIRST MEAN-VALUE THEOREM FOR INTEGRALS If f is continuous on [a,bl,then there is at least one number c in (a,b)for which f(x)dx f(c)(b-a). This number f(c)is called the average value (or mean value)of fon [a,b]. Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Mean-Value Theorems for Integrals
5.92) f(x)dx=(the average vahue of fon [a,b)(b-a). Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved