Skip to main content
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 5.1.49b
Chapter 5, Problem 5.1.49b

Sigma notation Evaluate the following expressions.                                                                                                                                          
(b)    10                                                                                                                                                                               
       ∑  (2κ + 1)                                                                                                                                                                          
       κ=1                         

Verified step by step guidance
1
Step 1: Understand the problem. The given expression is a summation in sigma notation: ∑ (2κ + 1), where κ starts at 1 and goes up to 10. This means we need to calculate the sum of the expression (2κ + 1) for each integer value of κ from 1 to 10.
Step 2: Write out the terms of the summation explicitly. Substitute κ = 1, κ = 2, ..., κ = 10 into the expression (2κ + 1). This will give the terms: (2*1 + 1), (2*2 + 1), ..., (2*10 + 1).
Step 3: Simplify each term. For example, when κ = 1, the term becomes 2*1 + 1 = 3. Similarly, calculate the terms for κ = 2, κ = 3, ..., κ = 10.
Step 4: Add all the simplified terms together. Once you have the values for each term, sum them up to find the total value of the summation.
Step 5: Verify your work. Double-check each substitution and addition to ensure accuracy in your calculations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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 the calculation of sums over a defined range.
Recommended video:
04:22
Sigma Notation

Series

A series is the sum of the terms of a sequence. In calculus, series can be finite or infinite, and they are often analyzed for convergence or divergence. Understanding how to manipulate and evaluate series is crucial for solving problems involving sigma notation, as it allows for the simplification and calculation of complex sums.
Recommended video:
06:00
Geometric Series

Index of Summation

The index of summation is the variable used to represent the position of each term in the sequence being summed. It typically starts at a specified value and increments by one until it reaches a defined upper limit. Mastery of how to correctly interpret and manipulate the index is essential for accurately evaluating expressions in sigma notation.
Recommended video:
04:22
Sigma Notation
Related Practice
Textbook Question

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .

(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2. 

1
views
Textbook Question

Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table.

(b) Find the midpoint Riemann sum approximation to the displacement on [0,2] with n = 2 and .n = 4 .

1
views
Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.


ƒ(x) = 4 - 2x on [0,4]


(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

1
views
Textbook Question

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

(b) 4 + 5 + 6 + 7 + 8 + 9

Textbook Question

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .

1
views
Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

f(𝓍) = x³ on [-1,2]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

1
views