Although not part of Theorem 7.4.2, it is true that the product of integrable functions is integrable. Provide the details for each step in the following proof of this fact: (a) If f satisfies |f(x)| < M on [a, 6], show|(f(x))? – (f(y))²| < 2M|f(x) – f(y)|.(b) Prove that if f is integrable on [a, b], then so is f².(c) Now show that if f and g are integrable, then fg is integrable. (Consider(f +g)².)

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(a) If f satisfies |f(x)| < M on [a, 6], show |(f(x))? – (f(y))²| < 2M|f(x) – f(y)|. (b) Prove that if f is integrable on [a, b], then so is f². (c) Now show that if f and g are integrable, then fg is integrable. (Consider (f +g)².)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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Question

Although not part of Theorem 7.4.2, it is true that the product of integrable functions is integrable. Provide the details for each step in the following proof of this fact:

(a) If f satisfies |f(x)| < M on [a, 6], show
|(f(x))? – (f(y))²| < 2M|f(x) – f(y)|.
(b) Prove that if f is integrable on [a, b], then so is f².
(c) Now show that if f and g are integrable, then fg is integrable. (Consider
(f +g)².)

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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