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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 5.2.79
Capitolo 5, Problema 5.2.79

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.


∫₀² (2𝓍 + 1) d𝓍

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Step 1: Recall the definition of the definite integral using Riemann sums. The definite integral ∫ₐᵇ f(𝓍) d𝓍 can be approximated by a sum: limₙ→∞ Σᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍, where Δ𝓍 = (b - a)/n and 𝓍ᵢ = a + iΔ𝓍 for right Riemann sums.
Step 2: Identify the given integral ∫₀² (2𝓍 + 1) d𝓍. Here, the interval [a, b] is [0, 2], the function f(𝓍) is 2𝓍 + 1, and we will use right Riemann sums.
Step 3: Compute Δ𝓍 = (b - a)/n = (2 - 0)/n = 2/n. This represents the width of each subinterval.
Step 4: Determine the right endpoints of the subintervals, 𝓍ᵢ = a + iΔ𝓍 = 0 + i(2/n) = (2i/n). Substitute these into the function f(𝓍): f(𝓍ᵢ) = 2(2i/n) + 1 = (4i/n) + 1.
Step 5: Write the Riemann sum approximation: Σᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍 = Σᵢ₌₁ⁿ [(4i/n) + 1](2/n). Expand and simplify the sum, then take the limit as n → ∞ to evaluate the definite integral.

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Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The value of the definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.
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Definition of the Definite Integral

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into rectangles. The sum of the areas of these rectangles, calculated using sample points (like right endpoints), provides an estimate of the integral. As the number of rectangles increases and their width decreases, the Riemann sum approaches the exact value of the definite integral.
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Introduction to Riemann Sums

Theorem 5.1 (Fundamental Theorem of Calculus)

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on [a, b], then the definite integral of its derivative over that interval equals the difference in the values of the function at the endpoints. This theorem provides a powerful tool for evaluating definite integrals and establishes the relationship between the two main branches of calculus.
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Fundamental Theorem of Calculus Part 1
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Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.


∫₁⁴ (𝓍²―1) d𝓍

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Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1― (𝓍/315)²) . Find the average height of the arch above the ground.

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Displacement from velocity The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0,t], where 0 ≤ t ≤ 3.

v(t) = { 30 if 0 ≤ t ≤ 2

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44 if 2.5 < t ≤ 3

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Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ sec² (10𝓍 + 7) d𝓍

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Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 ∫ sin² 𝓍 d𝓍

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Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  ∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)

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