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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 5.2.53d
Chapter 5, Problem 5.2.53d

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.
(a) βˆ«β‚€Β³ 5Ζ’(𝓍) d𝓍

Verified step by step guidance
1
Step 1: Recognize the property of integrals that allows constants to be factored out. Specifically, for any constant c and function f(x), βˆ«β‚α΅‡ cΖ’(𝓍) d𝓍 = c βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍.
Step 2: Apply this property to the given integral βˆ«β‚€Β³ 5Ζ’(𝓍) d𝓍. Here, the constant 5 can be factored out, resulting in 5 βˆ«β‚€Β³ Ζ’(𝓍) d𝓍.
Step 3: Substitute the value of βˆ«β‚€Β³ Ζ’(𝓍) d𝓍 provided in the problem, which is 2.
Step 4: Multiply the constant 5 by the value of the integral βˆ«β‚€Β³ Ζ’(𝓍) d𝓍 (which is 2) to complete the evaluation.
Step 5: The result of the integral βˆ«β‚€Β³ 5Ζ’(𝓍) d𝓍 is obtained by performing the multiplication in Step 4.

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Key Concepts

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

Properties of Integrals

The properties of integrals, particularly the linearity property, state that the integral of a constant multiplied by a function can be factored out. This means that ∫a^b kΖ’(𝓍) d𝓍 = k∫a^b Ζ’(𝓍) d𝓍, where k is a constant. This property simplifies the evaluation of integrals by allowing constants to be taken outside the integral.
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Properties of Functions

Definite Integrals

Definite integrals represent the signed area under a curve between two limits. The notation ∫a^b Ζ’(𝓍) d𝓍 indicates the integral of the function Ζ’(𝓍) from the lower limit a to the upper limit b. The result of a definite integral is a number that quantifies this area, which can be positive, negative, or zero depending on the function's behavior over the interval.
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Additivity of Integrals

The additivity property of integrals states that the integral over an interval can be split into the sum of integrals over subintervals. Specifically, ∫a^c Ζ’(𝓍) d𝓍 = ∫a^b Ζ’(𝓍) d𝓍 + ∫b^c Ζ’(𝓍) d𝓍 for any point b between a and c. This property is useful for evaluating integrals over larger intervals by breaking them down into smaller, manageable parts.
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Related Practice
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‚ƒβΆ (1―2𝓍) d𝓍 ; n = 6

Textbook Question

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) βˆ«β‚€^Ο€/¹⁢ sec Β² 4𝓍 d𝓍

Textbook Question

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

Ζ’(𝓍) = xΒ² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

Textbook Question

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

(d) If βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍 = βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍, then Ζ’ is a constant function. 

Textbook Question

Area functions The graph of Ζ’ is shown in the figure. Let A(x) = βˆ«β‚€Λ£ Ζ’(t) dt and F(x) = βˆ«β‚‚Λ£ Ζ’(t) dt be two area functions for Ζ’. Evaluate the following area functions.

(d) F(8)

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Textbook Question

Sigma notation Evaluate the following expressions.

(d)     5                                                                                                                                                                              

       βˆ‘ (1 + nΒ²)                                                                                                                                                                          

       n=1