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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)

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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.

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(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.


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

                                                                                                                                                                                     (d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then                                                                                                                                   

     B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

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

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