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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 5.1.47a
Chapter 5, Problem 5.1.47a

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(a) 1 + 2 + 3 + 4 + 5

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Step 1: Understand the problem. Sigma notation is a way to represent a sum of terms in a compact form using the Greek letter Σ. The goal is to express the sum 1 + 2 + 3 + 4 + 5 using this notation.
Step 2: Identify the pattern in the terms of the sum. The terms are consecutive integers starting from 1 and ending at 5. This suggests a simple arithmetic sequence.
Step 3: Define the general term of the sequence. In this case, the general term can be represented as 'k', where k is the index of summation.
Step 4: Determine the range of summation. The index k starts at 1 and ends at 5, so the range of summation is k = 1 to k = 5.
Step 5: Write the sum in sigma notation. Combine the general term and the range of summation to express the sum as: ∑kk=15

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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 numbers. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be added. The notation typically includes an index of summation, which specifies the starting and ending values for the variable that represents the terms in the sum.
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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

Finite Series

A finite series is the sum of a finite number of terms from a sequence. In the context of the given question, the series 1 + 2 + 3 + 4 + 5 is a finite series with five terms. Understanding finite series is essential for expressing sums in sigma notation, as it helps in determining the appropriate limits and the general term to be summed.
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Related Practice
Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(a) ∫₀¹ (4𝓍―2𝓍³) d𝓍

Textbook Question

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).

(a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .

ƒ(t) = 2t + 5 , a = 0

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Textbook Question

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

(a) Estimate the zeros of the area function A(𝓍) = ∫₀ˣ ƒ(t) dt , for 0 ≤ 𝓍 ≤ 10 .


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Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A (―2)

Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(a) ∫₀^π/2 (2 sin θ ― cos θ) dθ

Textbook Question

Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3t² + 1 on the interval 0 ≤ t ≤ 4, where t is measured in seconds.

(a) Divide the interval [0,4] into n = 4 subintervals, [0,1] , [1.2] , [2,3] , and [3,4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure)