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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 5.2.37
Chapter 5, Problem 5.2.37

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ’ on [a,b]. Identify Ζ’ and express the limit as a definite integral.                                
          n                                                                                                                                                                              
    lim   βˆ‘   π“*β‚– (ln 𝓍*β‚–) βˆ†π“β‚– on [1,2]                                                                                                                                                                            
  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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Step 1: Recognize the structure of the given limit. The expression lim βˆ‘ 𝓍ₖ (ln 𝓍ₖ) βˆ†π“β‚– as βˆ† β†’ 0 is a Riemann sum. A Riemann sum approximates the area under a curve by summing up small rectangles, and as the width of these rectangles (βˆ†π“β‚–) approaches zero, the sum converges to a definite integral.
Step 2: Identify the interval of integration. The problem specifies the interval [1, 2], which means the definite integral will be evaluated over this range.
Step 3: Determine the function Ζ’(𝓍) being integrated. In the Riemann sum, the term 𝓍ₖ (ln 𝓍ₖ) corresponds to the function Ζ’(𝓍) = 𝓍 ln(𝓍). This is the function that will be integrated.
Step 4: Write the definite integral. The limit of the Riemann sum can be expressed as the definite integral of Ζ’(𝓍) = 𝓍 ln(𝓍) over the interval [1, 2]. Using proper notation, this is written as: ∫a1b2x⁡ln⁡xdx
Step 5: Conclude the process. The definite integral ∫a1b2x⁡ln⁡xdx represents the limit of the given Riemann sum as βˆ† β†’ 0. This integral can now be evaluated using standard techniques of integration, if needed.

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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 by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function's value at that point is multiplied by the width of the subinterval. As the number of subintervals increases and their width decreases, the Riemann sum approaches the exact value of the definite integral.
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Introduction to Riemann Sums

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫[a,b] f(x) dx and can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals have numerous applications in calculating areas, volumes, and solving problems in physics and engineering.
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Definition of the Definite Integral

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. Limits are fundamental in defining both derivatives and integrals. In the context of Riemann sums, the limit is taken as the width of the subintervals approaches zero, allowing for the transition from a sum of areas of rectangles to the exact area under the curve, represented by the definite integral.
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Related Practice
Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


βˆ«β‚‹β‚‚β»ΒΉ 𝓍⁻³ d𝓍

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Textbook Question

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              

                                                                                                                                                                                        

 βˆ« (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

Textbook Question

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

                                                                                                                                                                              

 βˆ«β‚/₃^ΒΉ/√³ 4/(9𝓍² + 1) d𝓍

Textbook Question

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

                                                                                                                                                                    

 βˆ« d𝓍 / (√1 ― 9𝓍²)

Textbook Question

{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of Ζ’ and the 𝓍-axis on the given interval. Graph Ζ’ and show the region 𝑅.                                              

                                                                                                                                                                                    

 Ζ’(𝓍) = 𝓍² (𝓍 ― 2) on [ ―1 , 3]

Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  βˆ« d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)