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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.51d
Capitolo 5, Problema 5.3.51d

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)

Guida verificata passo dopo passo
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Step 1: Recognize that the integral ∫₀⁸ 3𝓍(4 ― 𝓍) d𝓍 can be split into two parts: ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 and ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍, based on the interval of integration.
Step 2: Use the given information that ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 = 32 to evaluate the first part of the integral.
Step 3: For the second part, ∫₄⁸ 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 ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍 directly by substitution or other methods.
Step 5: Combine the results of the two integrals, ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 and ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍, to find the value of ∫₀⁸ 3𝓍(4 ― 𝓍) d𝓍.

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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ₐᵇ 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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Definition of the Definite Integral

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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Properties of Functions

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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Euler's Method
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Domanda del libro di testo

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

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.                                                                                                                                                

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


ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4


d) Calculate the midpoint Riemann sum.

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Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

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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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