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5. The following limit is used to find the area y = f(x) from x = a to x = b. Evaluate the limit 2-√(21) ² 2i)² (20)]

5. The following limit is used to find the area y = f(x) from x = a to x = b. Evaluate the limit 2-√(21) ² 2i)² (20)]

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**5. The following limit is used to find the area under a curve** \( y = f(x) \) from \( x = a \) to \( x = b \). **Evaluate the limit** \[ \lim_{{n \to \infty}} \frac{2}{n} \sum_{{i=1}}^{n} \left[ \left( 1 + \frac{2i}{n} \right)^2 - \left( 1 + \frac{2i}{n} \right) \right] \] This expression represents a Riemann sum, which is a method for approximating the integral (or the area under a curve) over a specified interval. The limit of this sum as \( n \) approaches infinity gives the exact area. The terms inside the brackets involve evaluating a function at specific points and represent the height of rectangles under the curve, while \(\frac{2}{n}\) represents the width of these rectangles.
Transcribed Image Text:**5. The following limit is used to find the area under a curve** \( y = f(x) \) from \( x = a \) to \( x = b \). **Evaluate the limit** \[ \lim_{{n \to \infty}} \frac{2}{n} \sum_{{i=1}}^{n} \left[ \left( 1 + \frac{2i}{n} \right)^2 - \left( 1 + \frac{2i}{n} \right) \right] \] This expression represents a Riemann sum, which is a method for approximating the integral (or the area under a curve) over a specified interval. The limit of this sum as \( n \) approaches infinity gives the exact area. The terms inside the brackets involve evaluating a function at specific points and represent the height of rectangles under the curve, while \(\frac{2}{n}\) represents the width of these rectangles.
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