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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.2.32c
Capitolo 5, Problema 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.


∫₀² (𝓍²―2) d𝓍 ; n = 4

Guida verificata passo dopo passo
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Step 1: Understand the problem. You are tasked with approximating the definite integral ∫₀² (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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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 ∫ₐᵇ 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 Δx, 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
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{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


∫₁⁷ 1/𝓍 d𝓍 ; n = 6

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Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

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Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

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Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          

                                                                                                                                                                                     (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function. 

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Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

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