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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.51d
Chapter 5, Problem 5.3.51d

Properties of integrals Use only the fact that βˆ«β‚€β΄ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(d) βˆ«β‚€βΈ 3𝓍(4 ― 𝓍) d(𝓍)

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Step 1: Recognize that the integral βˆ«β‚€βΈ 3𝓍(4 ― 𝓍) d𝓍 can be split into two parts: βˆ«β‚€β΄ 3𝓍(4 ― 𝓍) d𝓍 and βˆ«β‚„βΈ 3𝓍(4 ― 𝓍) d𝓍, based on the interval of integration.
Step 2: Use the given information that βˆ«β‚€β΄ 3𝓍(4 ― 𝓍) d𝓍 = 32 to evaluate the first part of the integral.
Step 3: For the second part, βˆ«β‚„βΈ 3𝓍(4 ― 𝓍) d𝓍, consider the symmetry of the function 3𝓍(4 ― 𝓍). Analyze whether the function changes sign or remains symmetric over the interval [4, 8].
Step 4: If the function is symmetric and the integral over [0, 4] is known, use properties of symmetry to determine the integral over [4, 8]. Alternatively, compute βˆ«β‚„βΈ 3𝓍(4 ― 𝓍) d𝓍 directly by substitution or other methods.
Step 5: Combine the results of the two integrals, βˆ«β‚€β΄ 3𝓍(4 ― 𝓍) d𝓍 and βˆ«β‚„βΈ 3𝓍(4 ― 𝓍) d𝓍, to find the value of βˆ«β‚€βΈ 3𝓍(4 ― 𝓍) d𝓍.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as βˆ«β‚α΅‡ f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b].
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Properties of Integrals

The properties of integrals include linearity, additivity, and the ability to change limits. For instance, the integral of a sum is the sum of the integrals, and the integral from a to b can be expressed as the negative of the integral from b to a. These properties allow for simplification and manipulation of integrals to facilitate evaluation.
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Substitution Method

The substitution method is a technique used to simplify the evaluation of integrals by changing the variable of integration. By substituting a new variable, often denoted as u, the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler function.
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Related Practice
Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t β‰₯ 5, find the function that gives the displacement between t = 0 and any time t β‰₯ 5.

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Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} Ζ’(𝓍) = e Λ£/β‚‚ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums. 

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


βˆ«β‚β· 1/𝓍 d𝓍 ; n = 6

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (d) ∫ cos 𝓍/7 d𝓍

Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(d) βˆ«β‚†Β³ (Ζ’(𝓍) + 2g(𝓍)) d𝓍

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          

                                                                                                                                                                                     (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function.