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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 5.2.32c
Chapter 5, Problem 5.2.32c

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 
(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€Β² (𝓍²―2) d𝓍 ; n = 4

Verified step by step guidance
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Step 1: Understand the problem. You are tasked with approximating the definite integral βˆ«β‚€Β² (𝓍²―2) d𝓍 using left and right Riemann sums with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [0, 2] into n = 4 equal subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 0 and b = 2. Substitute the values to find Δ𝓍.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function f(𝓍) = 𝓍² - 2. The left endpoints are 𝓍₀ = 0, 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, and 𝓍₃ = 3Δ𝓍. Compute f(𝓍) at each left endpoint and multiply each value by Δ𝓍. Sum the results.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function f(𝓍) = 𝓍² - 2. The right endpoints are 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, 𝓍₃ = 3Δ𝓍, and 𝓍₄ = 4Δ𝓍. Compute f(𝓍) at each right endpoint and multiply each value by Δ𝓍. Sum the results.
Step 5: Compare the left and right Riemann sums to understand how the choice of endpoints affects the approximation of the integral. This comparison can provide insight into the accuracy of the method.

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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 value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on whether the left or right endpoints of the subintervals are used, the sums can yield different approximations, which converge to the actual integral as the number of rectangles increases.
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Introduction to Riemann Sums

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as βˆ«β‚α΅‡ f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be interpreted both geometrically, as the area between the curve and the x-axis, and analytically, as the limit of Riemann sums as the number of subdivisions approaches infinity.
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Definition of the Definite Integral

Subintervals and n

In the context of Riemann sums, 'n' refers to the number of subintervals into which the interval of integration is divided. Each subinterval has a width of Ξ”x, calculated as (b-a)/n. The choice of 'n' affects the accuracy of the approximation; a larger 'n' results in narrower rectangles and a more precise estimate of the integral, while a smaller 'n' may lead to greater error.
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Introduction to Riemann Sums
Related Practice
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(c)     4                                                                                                                                                                               

       βˆ‘ ΞΊΒ²                                                                                                                                                                          

       ΞΊ=1                         

Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(c) βˆ«β‚ƒβΆ (3Ζ’(𝓍) ― g(𝓍)) d𝓍

Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(c) 1Β² + 2Β² + 3Β² + 4Β²

Textbook Question

Zero net area Consider the function Ζ’(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function Ζ’(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is βˆ«β‚€α΅‡ Ζ’(𝓍) d𝓍 = 0 ? 

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of Ζ’(𝓍) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€^Ο€/2 cos 𝓍 d𝓍 ; n = 4

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