Textbook Question
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
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8. Definite Integrals
Riemann Sums
7:01 minutesProblem 5.R.11
Textbook Question
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.
β«βΒ² (πΒ²β4) dπ
Verified step by step guidance1
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: β«βα΅ f(x) dx = lim(nββ) Ξ£α΅’βββΏ f(xα΅’)Ξx, where Ξx = (b - a)/n and xα΅’ = a + iΞx.
Step 2: Identify the given function and interval. Here, f(x) = xΒ² - 4, a = 0, and b = 2. Calculate Ξx = (2 - 0)/n = 2/n, and the right endpoints xα΅’ = 0 + iΞx = 2i/n.
Step 3: Substitute f(xα΅’) and Ξx into the Riemann sum formula. The sum becomes Ξ£α΅’βββΏ [(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΅’βββΏ iΒ² = n(n+1)(2n+1)/6 and Ξ£α΅’βββΏ 1 = n. Substitute these formulas into the expression and simplify.
Step 5: Take the limit as n β 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 β«βΒ² (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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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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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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