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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 5.3.51c
Chapter 5, Problem 5.3.51c

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.


(c) βˆ«β‚„β° 6𝓍(4 ― 𝓍) d(𝓍)

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1
Step 1: Recognize that the integral given in the problem, βˆ«β‚„β° 6𝓍(4 ― 𝓍) d𝓍, is related to the integral βˆ«β‚€β΄ 3𝓍(4 ― 𝓍) d𝓍 = 32. Notice the limits of integration are reversed, and the integrand has been scaled by a factor of 2.
Step 2: Use the property of integrals that states reversing the limits of integration changes the sign of the integral. Specifically, βˆ«β‚α΅‡ f(𝓍) d𝓍 = -βˆ«α΅‡β‚ f(𝓍) d𝓍. Apply this property to rewrite βˆ«β‚„β° 6𝓍(4 ― 𝓍) d𝓍 as -βˆ«β‚€β΄ 6𝓍(4 ― 𝓍) d𝓍.
Step 3: Factor out the constant 6 from the integral using the property of integrals that allows constants to be factored out. This gives -6 βˆ«β‚€β΄ 𝓍(4 ― 𝓍) d𝓍.
Step 4: Recognize that βˆ«β‚€β΄ 𝓍(4 ― 𝓍) d𝓍 is equivalent to the given integral βˆ«β‚€β΄ 3𝓍(4 ― 𝓍) d𝓍 divided by 3, since the integrand in the given integral is scaled by a factor of 3. Therefore, βˆ«β‚€β΄ 𝓍(4 ― 𝓍) d𝓍 = 32 / 3.
Step 5: Substitute βˆ«β‚€β΄ 𝓍(4 ― 𝓍) d𝓍 = 32 / 3 into the expression -6 βˆ«β‚€β΄ 𝓍(4 ― 𝓍) d𝓍 to find the value of the integral. Simplify the expression to complete the solution.

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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]. Understanding this concept is crucial for evaluating integrals and applying properties related to limits.
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Definition of the Definite Integral

Properties of Integrals

The properties of integrals, such as linearity, additivity, and the reversal of limits, are essential for simplifying and evaluating integrals. For instance, the linearity property states that ∫(c * f(x)) dx = c * ∫f(x) dx for a constant 'c'. Additionally, the property of reversing limits states that βˆ«β‚α΅‡ f(x) dx = -βˆ«α΅‡β‚ f(x) dx. These properties allow for manipulation of integrals to facilitate easier computation.
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Substitution in Integrals

Substitution is a technique used to simplify the evaluation of integrals by changing the variable of integration. This method involves selecting a new variable 'u' that simplifies the integrand, allowing for easier integration. For example, if u = g(x), then dx can be expressed in terms of du, transforming the integral into a more manageable form. Mastery of substitution is vital for solving complex integrals effectively.
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Related Practice
Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

Ζ’(𝓍) = tan⁻¹ (3x - 1) on [0,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Textbook Question

Working with area functions Consider the function Ζ’ and its graph.

(c) Sketch a graph of A, for 0 ≀ 𝓍 ≀ 10 , without a scale on the y-axis.


Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = βˆ«β‚€^Ο€/2 (cos ΞΈ ― 2 sin ΞΈ) dΞΈ = ―1

(b) βˆ«β‚€^Ο€/2 (4 cos ΞΈ ― 8 sin ΞΈ) dΞΈ

Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(c) 1Β² + 2Β² + 3Β² + 4Β²

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’, Ζ’', and Ζ’'' are continuous functions for all real numbers.                                                                                                                                                           

                                                                                                                                                                    

(c) βˆ« sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

Textbook Question

Zero net area Consider the function Ζ’(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function Ζ’(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is βˆ«β‚€α΅‡ Ζ’(𝓍) d𝓍 = 0 ?