Textbook Question
Integration by Riemann sums Consider the integral β«ββ΄ (3πβ 2) dπ.
(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .
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8. Definite Integrals
Riemann Sums
6:09 minutesProblem 5.R.13
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«ββ΄ (πΒ³βπ) dπ
Verified step by step guidance1
Step 1: Recall the limit definition of the definite integral. The definite integral β«βα΅ f(π) dπ can be approximated using right Riemann sums with a regular partition. Divide the interval [a, b] into n subintervals of equal width Ξπ = (b - a)/n.
Step 2: For the given integral β«ββ΄ (πΒ³ - π) dπ, the interval [0, 4] is divided into n subintervals. The width of each subinterval is Ξπ = (4 - 0)/n = 4/n.
Step 3: The right endpoints of the subintervals are given by πα΅’ = a + iΞπ = 0 + i(4/n) = (4i)/n, where i ranges from 1 to n.
Step 4: Substitute the function f(π) = πΒ³ - π into the Riemann sum formula. The sum becomes Sβ = Ξ£α΅’βββΏ f(πα΅’)Ξπ = Ξ£α΅’βββΏ [(πα΅’Β³ - πα΅’) * Ξπ]. Replace πα΅’ with (4i)/n and Ξπ with 4/n.
Step 5: Simplify the expression for Sβ: Sβ = Ξ£α΅’βββΏ [((4i)/n)Β³ - (4i)/n] * (4/n). Expand and simplify the terms inside the summation. Then, take the limit as n β β to compute the exact value of the integral. Finally, use the Fundamental Theorem of Calculus to verify the result by directly evaluating β«ββ΄ (πΒ³ - π) dπ.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated as the limit of Riemann sums, which approximate the area by dividing the interval into smaller subintervals and summing the areas of rectangles formed. The notation β«βα΅ f(x) dx indicates the integral of f(x) from a to b.
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Riemann Sums
Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into rectangles. The height of each rectangle is determined by the function's value at specific points within each subinterval, such as the right endpoint, left endpoint, or midpoint. As the number of rectangles increases (and their width decreases), the Riemann sum approaches the exact value of the definite integral.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the definite integral of f from a to b can be computed as F(b) - F(a). This theorem provides a powerful way to evaluate definite integrals by finding an antiderivative rather than relying solely on Riemann sums.
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