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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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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
Textbook Question

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

(a) Graph ƒ on the interval 𝓍 ≥ 0.

Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

ƒ(𝓍) = cos 𝓍 ; a = 0 , b = π/2 , c = π

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

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

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.


(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.

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