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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 5.1.49c
Chapter 5, Problem 5.1.49c

Sigma notation Evaluate the following expressions.                                                                                                                                          
(c)     4                                                                                                                                                                               
       ∑ κ²                                                                                                                                                                          
       κ=1                         

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1
Step 1: Understand the problem. The given expression is a summation in sigma notation: ∑ κ², where κ starts at 1 and goes up to a specified upper limit. The summation involves squaring each value of κ and adding them together.
Step 2: Identify the range of summation. If the upper limit is not explicitly provided, assume it is given or needs clarification. For example, if the summation is ∑_{κ=1}^{n} κ², the range of κ is from 1 to n.
Step 3: Write out the terms of the summation explicitly. For example, if the upper limit is 4, the summation expands to κ² for κ = 1, 2, 3, and 4. This means you calculate 1² + 2² + 3² + 4².
Step 4: Use the formula for the sum of squares if the upper limit is large. The formula for the sum of squares of the first n integers is: S=n(n+1)(2n+1)6. This can simplify the calculation for large n.
Step 5: Substitute the values into the formula or manually compute the sum based on the range of κ. Ensure all terms are squared and added correctly to complete the summation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 representation of large sums.
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Sigma Notation

Summation of Squares

The summation of squares refers to the process of adding the squares of a sequence of integers. For example, the expression ∑ κ² from κ=1 to n represents the sum of the squares of the first n natural numbers. This concept is important in various mathematical contexts, including statistics and algebra, and has a well-known formula for its evaluation.
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Index of Summation

The index of summation is the variable used to represent the terms being summed in sigma notation. It typically starts at a specified lower limit and increments by one until it reaches an upper limit. Understanding how to manipulate the index is crucial for evaluating sums, especially when changing limits or transforming the expression being summed.
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Related Practice
Textbook Question

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.


(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(c) ∫₃⁶ (3ƒ(𝓍) ― g(𝓍)) 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.

(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

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Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(c) 1² + 2² + 3² + 4²

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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

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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