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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 5.1.31d
Capitolo 5, Problema 5.1.31d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums. 

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1
Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(𝓍) = e^(𝓍/2) over the interval [1, 4] with n = 6 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [1, 4] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / 6 = 3 / 6 = 0.5.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function. The left endpoints are: 𝓍₀ = 1, 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, and 𝓍₅ = 3.5. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function. The right endpoints are: 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, 𝓍₅ = 3.5, and 𝓍₆ = 4. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.
Step 5: Multiply each function value by the width of the subinterval, Δ𝓍 = 0.5, and sum the results for both the left and right Riemann sums. The left Riemann sum is Σ[ƒ(𝓍ᵢ) * Δ𝓍] for i = 0 to 5, and the right Riemann sum is Σ[ƒ(𝓍ᵢ) * Δ𝓍] for i = 1 to 6.

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Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over a specified interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and then summing the products of these values and the widths of the subintervals. The left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints.
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Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫[a,b] f(x) dx, where [a,b] is the interval of integration.
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Definition of the Definite Integral

Function Evaluation

Function evaluation involves substituting specific values into a function to determine its output. In the context of Riemann sums, evaluating the function at the endpoints of the subintervals is crucial for calculating the sum. For the given function f(x) = e^(x/2), evaluating it at the left and right endpoints of each subinterval will provide the necessary function values to compute the left and right Riemann sums.
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Evaluating Composed Functions
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