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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 5.2.35
Chapter 5, Problem 5.2.35

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‘ (𝓍ₖ*Β² + 1) βˆ†π“β‚– on [0,2]                                                                                                                                                                            
  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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1
Step 1: Recognize the structure of the given limit. The expression lim βˆ‘ (𝓍ₖ² + 1) βˆ†π“β‚– as βˆ† β†’ 0 represents a Riemann sum. A Riemann sum approximates the area under a curve by summing up the areas of rectangles, where the height of each rectangle is determined by the function value at a specific point within the interval.
Step 2: Identify the function Ζ’(𝓍) from the given Riemann sum. The term (𝓍ₖ² + 1) corresponds to the function Ζ’(𝓍) = 𝓍² + 1. This is the function being integrated over the interval [0, 2].
Step 3: Determine the interval of integration. The problem specifies the interval [0, 2], which means the definite integral will be evaluated from 𝓍 = 0 to 𝓍 = 2.
Step 4: Express the limit as a definite integral. The limit of the Riemann sum as βˆ† β†’ 0 is equivalent to the definite integral of the function Ζ’(𝓍) = 𝓍² + 1 over the interval [0, 2]. Using integral notation, this is written as: ∫a0b2x2+1dx
Step 5: Conclude the process. The definite integral ∫a0b2x2+1dx represents the exact area under the curve Ζ’(𝓍) = 𝓍² + 1 from 𝓍 = 0 to 𝓍 = 2.

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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. When evaluating Riemann sums, the limit is taken as the width of the subintervals approaches zero, which allows for the transition from a sum of areas of rectangles to the exact area under the curve. Understanding limits is crucial for grasping the foundational concepts of continuity, derivatives, and integrals.
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Related Practice
Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for Ζ’(𝓍)) = x + 1 on [0, 4] with n = 50.

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Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dt βˆ«β‚€α΅— d𝓍/(1 + 𝓍²) + βˆ«β‚ΒΉ/α΅— dx/(1 + 𝓍²)

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Textbook Question

Integrals with sinΒ² 𝓍 and cosΒ² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 βˆ« 𝓍 cos²𝓍² d𝓍

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Textbook Question

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

                                                                                                                                                                              

 βˆ«β‚‹β‚ΒΉ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

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Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.


βˆ«α΅ƒβ‚‹β‚ Ζ’(g(𝓍)) 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

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