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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 5.2.58b
Chapter 5, Problem 5.2.58b

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 
I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1
(b) ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ

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1
Step 1: Recognize that the given integral (b) ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ can be expressed in terms of the original integral I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1.
Step 2: Factor out the constant multiplier from the integrand in (b). Using the property of integrals, ∫ₐᵇ k * f(x) dx = k * ∫ₐᵇ f(x) dx, rewrite the integral as ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ = 4 * ∫₀^π/2 (cos θ ― 2 sin θ) dθ.
Step 3: Substitute the value of the original integral I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1 into the expression derived in Step 2.
Step 4: Multiply the constant factor (4) by the value of the original integral (―1) to simplify the expression.
Step 5: The result of the multiplication gives the value of the integral (b).

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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, such as linearity, allow us to manipulate integrals in useful ways. For instance, the integral of a sum can be expressed as the sum of the integrals, and constants can be factored out. This is crucial for evaluating integrals efficiently, especially when they can be expressed in terms of known integrals.
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Properties of Functions

Definite Integrals

Definite integrals represent the signed area under a curve between two limits. In this case, the limits are from 0 to π/2. Understanding how to compute definite integrals and their properties is essential for evaluating integrals like the one given in the question, as it provides the numerical value of the area.
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Definition of the Definite Integral

Substitution in Integrals

Substitution is a technique used to simplify the evaluation of integrals by changing the variable of integration. This method can transform a complex integral into a simpler form, making it easier to compute. Recognizing when and how to apply substitution is key to solving integrals effectively.
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Related Practice
Textbook Question

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral) 

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Textbook Question

Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       

(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍

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Textbook Question

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

                                                                                                                                                                    

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

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Textbook Question

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(b) Verify that .A'(𝓍) = ƒ(𝓍)

                                                                                                                                                            

ƒ(t) = 5 , a = -5

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Textbook Question

Generalizing the Mean Value Theorem for Integrals Suppose ƒ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) ∫ₐˣ ƒ(t) dt + (𝓍―a) ∫ₓᵇg(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that ∫ₐᶜ ƒ(t) dt = ƒ(c) (b ― c)                                                                                                              

                                                                                                                                                                                

(Source: The College Mathematics Journal, 33, 5, Nov 2002)

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Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

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