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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 5.1.47c
Chapter 5, Problem 5.1.47c

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(c) 1² + 2² + 3² + 4²

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Identify the pattern in the given sum: The terms are squares of consecutive integers starting from 1 up to 4. This suggests the general term is n², where n represents the integer.
Determine the range of the index variable: The integers in the sum start at 1 and end at 4. Therefore, the index variable n will range from 1 to 4.
Write the sum in sigma notation: Use the summation symbol (∑) to represent the sum. The general term n² will be placed after the summation symbol, and the limits of summation will be specified as n = 1 to n = 4.
Combine the elements: The sigma notation for the sum is ∑n1n2, where n starts at 1 and ends at 4.
Verify the representation: Ensure that the sigma notation correctly represents the original sum by expanding the terms and confirming they match 1² + 2² + 3² + 4².

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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 typically includes an index of summation, which specifies the starting and ending values for the variable that represents the terms.
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Sigma Notation

Index of Summation

The index of summation is a variable used in sigma notation to denote the position of each term in the sequence being summed. It usually starts at a specified lower limit and increments by one until it reaches an upper limit. For example, in the sum Σ from i=1 to n, 'i' is the index that takes on values from 1 to n, allowing for the systematic addition of terms.
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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. In the context of the given question, the sum 1² + 2² + 3² + 4² can be expressed in sigma notation as Σ from i=1 to 4 of i². This concept is important in various mathematical applications, including statistics and algebra, where the properties of squared terms are frequently utilized.
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Completing the Square to Rewrite the Integrand
Related Practice
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(c)     4                                                                                                                                                                               

       ∑ κ²                                                                                                                                                                          

       κ=1                         

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)

Textbook Question

Working with area functions Consider the function ƒ and its graph.

(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.


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 .

Textbook Question

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ? 

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


∫₀² (𝓍²―2) d𝓍 ; n = 4