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 5.2.79

Chapter 5, Problem 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

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: 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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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 two main branches of calculus.

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