Textbook Question
Working with area functions Consider the function Ζ and its graph.
(b) Estimate the points (if any) at which A has a local maximum or minimum.
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.1.51b
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
Ζ(π) = 3 βx on [0,4] ; n = 40
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with estimating the area under the curve of the function Ζ(π) = 3βx on the interval [0,4] using Riemann sums with n = 40 subintervals. This involves dividing the interval into smaller subintervals and summing the areas of rectangles under the curve.
Step 2: Divide the interval [0,4] into n = 40 subintervals. The width of each subinterval, Ξx, is calculated as Ξx = (b - a) / n, where a = 0 and b = 4. Substitute these values into the formula to find Ξx.
Step 3: Determine the x-values for the endpoints of each subinterval. These x-values are given by xβ, xβ, xβ, ..., xβ, where xβ = a and xβ = b. Use the formula xα΅’ = a + iΞx (for i = 0, 1, 2, ..., n) to calculate the x-values.
Step 4: Evaluate the function Ζ(π) = 3βx at the appropriate x-values. Depending on whether you are using left Riemann sums, right Riemann sums, or midpoint Riemann sums, evaluate Ζ(π) at the left endpoints, right endpoints, or midpoints of each subinterval.
Step 5: Compute the Riemann sum. Multiply the function values by the width of the subintervals (Ξx) and sum them up. The formula for the Riemann sum is Ξ£[Ζ(xα΅’)Ξx], where the summation runs over all subintervals. This sum provides an approximation of the area under the curve.
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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 area under a curve by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function value at that point is multiplied by the width of the subinterval. The sum of these products gives an estimate of the total area. As the number of subintervals (n) increases, the approximation becomes more accurate.
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06:11
Introduction to Riemann Sums
Definite Integrals
Definite integrals represent the exact area under a curve between two points on the x-axis. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral of a function over an interval provides a precise value for the area, contrasting with Riemann sums, which provide an approximation that improves with larger n.
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05:43Definition of the Definite Integral
Limit of Riemann Sums
The limit of Riemann sums as the number of subintervals approaches infinity leads to the exact value of the definite integral. This concept is crucial in calculus, as it formalizes the transition from discrete approximations to continuous areas. Understanding this limit helps in grasping how integration works and why it is a fundamental tool for calculating areas under curves.
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06:11Introduction to Riemann Sums
Related Practice
Textbook Question
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
β«βΒΉ (πΒ² + 1) dπ
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Textbook Question
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(b) β«ββΆ (β3g(π)) dπ
Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(b) Graph Ζ and A.
Ζ(π) = 1/π ; a = 1 , b = 4 , c = 6
Textbook Question
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(b) β«ββΆ (f(π) β g(π)) dπ
Textbook Question
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
β«βΒΉ cos β»ΒΉ π dπ