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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 5.1.49a
Chapter 5, Problem 5.1.49a

Sigma notation Evaluate the following expressions.
(a)    10                                                                                                                                                                               
       ∑ κ                                                                                                                                                                          
       κ=1                         

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1
Step 1: Understand the problem. The given expression involves sigma notation, which represents the summation of terms. The general form of sigma notation is ∑_{k=a}^{b} f(k), where 'k' is the index of summation, 'a' is the lower limit, 'b' is the upper limit, and f(k) is the function to be summed.
Step 2: Identify the components of the given sigma notation. In this case, the summation is ∑_{k=1}^{10} k, which means we are summing the values of 'k' from 1 to 10.
Step 3: Write out the terms of the summation explicitly. Substitute the values of 'k' from 1 to 10 into the expression 'k'. This gives: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
Step 4: Recognize that this is an arithmetic series. The sum of the first 'n' natural numbers can be calculated using the formula S = n(n+1)/2, where 'n' is the largest number in the series.
Step 5: Apply the formula for the sum of the first 'n' natural numbers. Substitute n = 10 into the formula S = n(n+1)/2 to find the sum of the series. This will give the final result.

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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 limits that specify the starting and ending indices of the summation, allowing for efficient representation of large sums.
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Sigma Notation

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. It is usually denoted by a letter, such as 'k', and takes on integer values from a specified lower limit to an upper limit. Understanding how to manipulate and evaluate the index is crucial for correctly calculating the sum represented by sigma notation.
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Evaluating Series

Evaluating series involves calculating the total sum of the terms defined by the sigma notation. This process may require substituting values for the index of summation, performing arithmetic operations, and sometimes applying formulas for known series. Mastery of techniques for evaluating series is essential for solving problems that involve sigma notation.
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Related Practice
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Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(a) Describe the motion of the object over the interval [0,6].

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(a) Graph ƒ on the interval 𝓍 ≥ 0.

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Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .


(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

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Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earth’s surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)

(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints x₀ = 34 , x₁ = 40 , x₂ = 46 , x₃ = 52 , x₄ = 58 , and x₅ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.

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Textbook Question

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.


(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)                                                                                                             

                                                                                                                                                                                                

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Textbook Question

Area functions for constant 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 for ƒ.

ƒ(t) = 5 , a = 0

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