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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.16d
Capitolo 5, Problema 5.5.16d

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    
 (d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

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Step 1: Recognize the integral ∫₀^(π/16) sec²(4𝓍) d𝓍 as a standard form integral. The integral of sec²(u) with respect to u is tan(u).
Step 2: Identify the substitution needed for the integral. Let u = 4𝓍, which implies that du = 4 d𝓍. Rewrite the integral in terms of u.
Step 3: Adjust the limits of integration based on the substitution. When 𝓍 = 0, u = 4(0) = 0. When 𝓍 = π/16, u = 4(π/16) = π/4.
Step 4: Rewrite the integral using the substitution: ∫₀^(π/16) sec²(4𝓍) d𝓍 becomes (1/4) ∫₀^(π/4) sec²(u) du, where the factor 1/4 comes from the substitution du = 4 d𝓍.
Step 5: Evaluate the integral ∫₀^(π/4) sec²(u) du using the antiderivative of sec²(u), which is tan(u). Substitute the limits of integration into tan(u) and multiply by the factor 1/4 to complete the solution.

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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b]. Understanding how to evaluate definite integrals is crucial for solving problems in calculus.
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Definition of the Definite Integral

Integration Techniques

Integration techniques are methods used to find the integral of a function, especially when it cannot be integrated using basic formulas. Common techniques include substitution, integration by parts, and using trigonometric identities. In this context, recognizing the appropriate technique to apply is essential for evaluating the integral effectively, particularly when dealing with functions like sec²(4x).
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Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as secant (sec), are fundamental in calculus and relate angles to ratios of sides in right triangles. The secant function is defined as the reciprocal of the cosine function, sec(x) = 1/cos(x). Understanding the properties and derivatives of trigonometric functions is vital for evaluating integrals involving these functions, as they often appear in various calculus problems.
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Domanda del libro di testo

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

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Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

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


{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

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{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..


∫₀² (𝓍²―2) d𝓍 ; n = 4

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


ƒ(𝓍) = 1/x on [1,6] ; n = 5


(d) Calculate the midpoint Riemann sum.

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