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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.R.11
Chapter 5, Problem 5.R.11

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 


βˆ«β‚€Β² (𝓍²―4) d𝓍

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1
Step 1: Recall the limit definition of the definite integral. For a function f(x) over the interval [a, b], the definite integral can be approximated using right Riemann sums: βˆ«β‚α΅‡ f(x) dx = lim(nβ†’βˆž) Ξ£α΅’β‚Œβ‚βΏ f(xα΅’)Ξ”x, where Ξ”x = (b - a)/n and xα΅’ = a + iΞ”x.
Step 2: Identify the given function and interval. Here, f(x) = xΒ² - 4, a = 0, and b = 2. Calculate Ξ”x = (2 - 0)/n = 2/n, and the right endpoints xα΅’ = 0 + iΞ”x = 2i/n.
Step 3: Substitute f(xα΅’) and Ξ”x into the Riemann sum formula. The sum becomes Ξ£α΅’β‚Œβ‚βΏ [(2i/n)Β² - 4](2/n). Expand and simplify the terms inside the summation.
Step 4: Use summation formulas to evaluate the sum. For example, Ξ£α΅’β‚Œβ‚βΏ iΒ² = n(n+1)(2n+1)/6 and Ξ£α΅’β‚Œβ‚βΏ 1 = n. Substitute these formulas into the expression and simplify.
Step 5: Take the limit as n β†’ ∞. Simplify the resulting expression to find the value of the definite integral. Finally, use the Fundamental Theorem of Calculus to verify the result by directly evaluating βˆ«β‚€Β² (xΒ² - 4) dx.

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Key Concepts

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

Limit Definition of the Definite Integral

The limit definition of the definite integral involves approximating the area under a curve using Riemann sums. As the number of subintervals increases and their width approaches zero, the sum of the areas of rectangles formed under the curve converges to the exact area, which is represented by the integral. This concept is foundational for understanding how integrals are derived and calculated.
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Definition of the Definite Integral

Riemann Sums

Riemann sums are a method for approximating the total area under a curve by dividing the interval into smaller subintervals and summing the areas of rectangles formed. In the context of the limit definition, right Riemann sums use the right endpoint of each subinterval to determine the height of the rectangles. This approach is crucial for evaluating definite integrals and understanding their geometric interpretation.
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Introduction to Riemann Sums

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the difference in the values of the original function at the endpoints. This theorem provides a powerful tool for evaluating definite integrals and verifying results obtained through the limit definition and Riemann sums.
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Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question

Area functions and the Fundamental Theorem Consider the function

Ζ’(t) = { t      if  β€•2 β‰€ t < 0

tΒ²/2    if    0 β‰€ t β‰€ 2

and its graph shown below. Let F(𝓍) = βˆ«β‚‹β‚Λ£ Ζ’(t) dt and G(𝓍) = βˆ«β‚‹β‚‚Λ£ Ζ’(t) dt.

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for β€•2 β‰€ π“ < 0.

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Textbook Question

Use geometry and properties of integrals to evaluate the following definite integrals.                                                                                          

                                                                                                                                                                       

 βˆ«β‚€β΄ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

98
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Textbook Question

Function defined by an integral Let H (𝓍) = βˆ«β‚€Λ£ √(4 ― tΒ²) dt, for ― 2 ≀ 𝓍 ≀ 2.

(e) Find the value of s such that H (𝓍) = sH(―𝓍)

83
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Textbook Question

Integration by Riemann sums Consider the integral βˆ«β‚β΄ (3𝓍― 2) d𝓍.


(c) Evaluate the definite integral by taking the limit as n β†’βˆž of the Riemann sum in part (b).

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’ and Ζ’' are continuous functions for all real numbers.

(b) Given an area function A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt and an antiderivative F of Ζ’, it follows that A'(𝓍) = F(𝓍) .

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Textbook Question

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

                                                                                                                                                                    

 βˆ« (cos 7Ο‰) /(16 + sinΒ² 7Ο‰) dΟ‰ 

71
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