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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.2.33c
Capitolo 5, Problema 5.2.33c

{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.
∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Guida verificata passo dopo passo
1
Step 1: Understand the problem. You are tasked with approximating the definite integral ∫₁⁷ (1/𝓍) d𝓍 using left and right Riemann sums with n = 6 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [1, 7] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (7 - 1) / 6 = 6 / 6 = 1.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function f(𝓍) = 1/𝓍. The left endpoints are 𝓍₀ = 1, 𝓍₁ = 2, 𝓍₂ = 3, 𝓍₃ = 4, 𝓍₄ = 5, and 𝓍₅ = 6. Compute the sum: Left Riemann Sum = Δ𝓍 * [f(𝓍₀) + f(𝓍₁) + f(𝓍₂) + f(𝓍₃) + f(𝓍₄) + f(𝓍₅)].
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function f(𝓍) = 1/𝓍. The right endpoints are 𝓍₁ = 2, 𝓍₂ = 3, 𝓍₃ = 4, 𝓍₄ = 5, 𝓍₅ = 6, and 𝓍₆ = 7. Compute the sum: Right Riemann Sum = Δ𝓍 * [f(𝓍₁) + f(𝓍₂) + f(𝓍₃) + f(𝓍₄) + f(𝓍₅) + f(𝓍₆)].
Step 5: After setting up the sums, substitute the values of f(𝓍) = 1/𝓍 for each endpoint into the formulas for the left and right Riemann sums. Simplify the expressions to approximate the integral.

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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 between two specified limits, often denoted as ∫ₐᵇ f(x) dx. It quantifies the accumulation of quantities, such as area, over an interval [a, b]. The Fundamental Theorem of Calculus connects differentiation and integration, stating that the definite integral can be evaluated using the antiderivative of the function.
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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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