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 5.1.65
Chapter 5, Problem 5.1.65

Identifying Riemann sums Fill in the blanks with an interval and a value of n.


4
∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with
k = 1
n = ________ .

Verified step by step guidance
1
Step 1: Understand the structure of the Riemann sum. A Riemann sum approximates the area under a curve by summing up the areas of rectangles. The given sum is a right Riemann sum, which means the function values are evaluated at the right endpoints of the subintervals.
Step 2: Analyze the given sum. The expression ƒ(1 + k) • 1 suggests that the interval starts at 1, and the width of each subinterval (Δx) is 1. The term 'k' represents the index of the subintervals.
Step 3: Determine the interval. Since the sum starts at k = 1 and increments by 1 for each term, the interval can be determined by the range of values for x. The interval is [1, 1 + n], where n is the number of subintervals.
Step 4: Identify the value of n. The summation symbol (∑) indicates that there are 4 terms in the sum, which means n = 4. This corresponds to the number of subintervals used in the approximation.
Step 5: Fill in the blanks. Based on the analysis, the interval is [1, 5] (since n = 4 and the interval ends at 1 + n), and the value of n is 4.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and summing the products of these values and the widths of the subintervals. The choice of points (left, right, or midpoint) affects the approximation's accuracy.
Recommended video:
06:11
Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. This concept is fundamental in calculus, linking the idea of accumulation with the geometric interpretation of area.
Recommended video:
05:43
Definition of the Definite Integral

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It specifies the lower and upper bounds of the interval, using brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. Understanding interval notation is crucial for correctly identifying the limits of integration when working with Riemann sums and definite integrals.
Recommended video:
04:22
Sigma Notation
Related Practice
Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 ∫₀¹ 2e²ˣ d𝓍

Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 ∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍

Textbook Question

Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.


ƒ(𝓍) = sin 𝓍 on [―π/4, 3π/4]

Textbook Question

Is x¹² an even or odd function? Is sin x² an even or odd function?

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx