Explanation Given: • Integral: ∫ r ⋅ 5 − 4 − r 2 0 2 ⋅ d r Key Concepts Used: Definite Integral: The definite integral of f ( x ) over [ a , b ] is denoted by the integral sign ∫ f ( x ) ⋅ d x a b . It is often called the integral of f ( x ) over [ a , b ] . The process of computing integrals is called Integration. The function f ( x ) is called the integrand. The endpoints a a n d b of [ a , b ] are called the Limits of Integration. Substitution Method: When the integrand has the form f ( u ( x ) ) ⋅ u ' ( x ) , we can use the equation d u = u ' ( x ) ⋅ d x to rewrite the entire integral (including the d x term) in terms of u and it’s differential d u as follows: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u . U-Substitution is a technique we use when the integrand is a composite function. A composite function is a function inside another function of the form f ( u ( x ) ) where f ( x ) is the outside function and u ( x ) is the inside function. The first thing we will need to do is to recognize that we are being asked to integrate a product of a function and its derivative, and it takes the form of a composite function. This can be generalized for any product that is a composite function multiplied by the derivative of the inner function. Here in this problem we have to find the value of the given integral ∫ r ⋅ 5 − 4 − r 2 0 2 ⋅ d r . To find the value of the given integral, we have to use substitution process. Here we can substitute u = 5 − 4 − r 2 . Then the given integral becomes in terms of u . We have to integrate with respect to u . By applying the new limits of u which we found by calculation, we obtain the final solution. Calculation: Description: ∫ f ( u ( x ) ) ⋅ u ' ( x ) ⋅ d x = ∫ f ( u ) ⋅ d u Step 1 :Write the provided integral for which we have to find the value. ∫ r ⋅ 5 − 4 − r 2 0 2 ⋅ d r Step 2 : Write the value of u which we consider to substitute. u = 5 − 4 − r 2 Step 3 : Find the value of d u for the considered value of u and also find the limits of u . ∫ r ⋅ 5 − 4 − r 2 0 2 ⋅ d r L e t u = 5 − 4 − r 2 d u d r = d d r ( 5 − 4 − r 2 ) = 0 − 1 2 ⋅ 4 − r 2 ( − 2 r ) d u d r = − ( − 2 r ) 2 ⋅ 4 − r 2 = 2 r 2 ⋅ 4 − r 2 = r 4 − r 2 d u = r 4 − r 2 ⋅ d r A s a s s u m e d u = 5 − 4 − r 2 , − 4 − r 2 = u − 5 4 − r 2 = 5 − u T h e r e b y d u = r ⋅ d r 5 − u W h e n r = 0 , u = 5 − 4 − 0 2 = 5 − 2 = 3 W h e n r = 2 , u = 5 − 4 − 2 2 = 5 − 0 = 5 Calculation Continued: Step 4 : Substitute the values of u a n d d u and the limits of u obtained from the above step in the given integral to find its value...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 5.7, Problem 85E

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The value of the given integral

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Chapter 5.7, Problem 84E

Chapter 5.7, Problem 86E

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Chapter 5 Solutions

CALCULUS (CLOTH)

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

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The value of the given integral

Solution Summary: The author explains that the definite integral of f (x) over [a, b] is denoted by the integral sign. The process of computing integrals is called Integration.

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Chapter 5 Solutions