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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 5.1.29d
Capitolo 5, Problema 5.1.29d

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. 

Guida verificata passo dopo passo
1
Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(𝓍) = x² - 1 over the interval [2,4] with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [2,4] into n = 4 equal subintervals. The width of each subinterval, Δx, is calculated as Δx = (4 - 2) / 4 = 0.5.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function ƒ(𝓍). The left endpoints are: 𝓍₀ = 2, 𝓍₁ = 2.5, 𝓍₂ = 3, and 𝓍₃ = 3.5. Compute the function values ƒ(𝓍₀), ƒ(𝓍₁), ƒ(𝓍₂), and ƒ(𝓍₃).
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function ƒ(𝓍). The right endpoints are: 𝓍₁ = 2.5, 𝓍₂ = 3, 𝓍₃ = 3.5, and 𝓍₄ = 4. Compute the function values ƒ(𝓍₁), ƒ(𝓍₂), ƒ(𝓍₃), and ƒ(𝓍₄).
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 3, and the right Riemann sum is Σ[ƒ(𝓍ᵢ) * Δx] for i = 1 to 4.

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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 (left endpoints, right endpoints, or midpoints), and summing the products of these values and the widths of the subintervals. This technique helps in understanding the area under a curve.
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Introduction to Riemann Sums

Left and Right Riemann Sums

Left and right Riemann sums are specific types of Riemann sums that use the leftmost and rightmost points of each subinterval, respectively, to estimate the area under a curve. For the left Riemann sum, the function's value at the left endpoint of each subinterval is used, while for the right Riemann sum, the value at the right endpoint is used. These sums provide different approximations of the integral, which can be compared for accuracy.
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Left, Right, & Midpoint Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the exact area under the curve of the function between two points. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. Understanding the relationship between Riemann sums and definite integrals is crucial for grasping how these approximations converge to the actual area as the partition of the interval becomes finer.
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Definition of the Definite Integral
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Domanda del libro di testo

{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.


∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

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


{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

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Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(8)

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


ƒ(𝓍) = 1/x on [1,6] ; n = 5


(d) Calculate the midpoint Riemann sum.

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