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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 5.1.49f
Capitolo 5, Problema 5.1.49f

Sigma notation Evaluate the following expressions.                                                                                                                                          
  (f)      3                                                                                                                                                                               
       ∑ (3j ― 4)                                                                                                                                                                          
      j =1                         

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1
Step 1: Understand the problem. The given expression involves sigma notation, which represents the summation of a sequence. The formula provided is ∑(3j - 4), where j starts at 1 and goes up to a certain upper limit (not specified in the problem).
Step 2: Identify the summation formula. The general form of sigma notation is ∑(expression) from j = lower limit to upper limit. Here, the expression is (3j - 4), and the summation will involve substituting values of j starting from 1 up to the upper limit.
Step 3: Substitute values of j into the expression. For each value of j, calculate (3j - 4). For example, when j = 1, the term becomes (3 * 1 - 4). When j = 2, the term becomes (3 * 2 - 4), and so on.
Step 4: Add the results of each term. After calculating the individual terms for each value of j, sum them together to find the total value of the summation.
Step 5: If the upper limit is provided, repeat the substitution and addition process until j reaches the upper limit. If the upper limit is not specified, clarify the problem to determine the range of summation.

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Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be summed. The notation includes limits that specify the starting and ending indices of the summation, allowing for efficient calculation of series.
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Sigma Notation

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. In the context of sigma notation, if the expression being summed represents an arithmetic sequence, it can be simplified using formulas for the sum of such sequences, making calculations easier.
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Arithmetic Sequences - General Formula

Evaluating Series

Evaluating series involves calculating the total sum of the terms defined by the sigma notation. This process may require substituting the index values into the expression, computing each term, and then summing them up. Understanding how to manipulate and simplify the expression is crucial for efficient evaluation, especially for larger series.
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Geometric Series
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Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)  

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Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       ∑  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

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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.


(f) ∫₄¹ 2f(𝓍) d𝓍

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Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (f) ∫ d𝓍/√36 ―𝓍²

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