Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.2.83

Chapter 5, Problem 5.2.83

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

Verified step by step guidance

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: lim(n→∞) Σᵢ₌₁ⁿ f(xᵢ)Δx, where Δx = (b - a)/n and xᵢ = a + iΔx for right Riemann sums.

Step 2: Identify the function f(x) = x² - 1, the interval [1, 4], and the number of subintervals n. Calculate Δx = (4 - 1)/n = 3/n.

Step 3: Determine the sample points xᵢ for the right Riemann sum. For i = 1, 2, ..., n, xᵢ = 1 + iΔx = 1 + i(3/n).

Step 4: Substitute f(xᵢ) and Δx into the Riemann sum formula. The sum becomes Σᵢ₌₁ⁿ [(xᵢ² - 1) * Δx], where xᵢ = 1 + i(3/n) and Δx = 3/n.

Step 5: Simplify the expression for the sum and take the limit as n → ∞ to evaluate the definite integral. This involves expanding (1 + i(3/n))², simplifying the summation, and using summation formulas for i and i².

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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'.

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.

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 antiderivative and the area under the curve.

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Related Practice

Textbook Question

Use symmetry to explain why.

∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .

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Textbook Question

Variations on the substitution method Evaluate the following integrals.

∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵

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Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.

∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)

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Textbook Question

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.

469

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Textbook Question

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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Textbook Question

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

50 if 2 < t < 2.5

44 if 2.5 < t ≤ 3

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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

Verified video answer for a similar problem:

Key Concepts

Definite Integral

Riemann Sums

Theorem 5.1 (Fundamental Theorem of Calculus)

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