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 5.1.11

Chapter 5, Problem 5.1.11

Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length ∆𝓍? List the grid points x₀ , x₁ , x₂ , x₃ and x₄. Which points are used for the left, right, and midpoint Riemann sums?

Verified step by step guidance

Step 1: To find the subinterval length (∆𝓍), use the formula ∆𝓍 = (b - a) / n, where [a, b] is the interval and n is the number of subintervals. Here, a = 1, b = 3, and n = 4.

Step 2: Calculate the grid points x₀, x₁, x₂, x₃, and x₄. Start with x₀ = a (the left endpoint of the interval), and then add ∆𝓍 successively to find the remaining points: x₁ = x₀ + ∆𝓍, x₂ = x₁ + ∆𝓍, x₃ = x₂ + ∆𝓍, and x₄ = x₃ + ∆𝓍.

Step 3: For the left Riemann sum, use the grid points x₀, x₁, x₂, and x₃ as the sample points. These are the left endpoints of each subinterval.

Step 4: For the right Riemann sum, use the grid points x₁, x₂, x₃, and x₄ as the sample points. These are the right endpoints of each subinterval.

Step 5: For the midpoint Riemann sum, calculate the midpoints of each subinterval. The midpoints are given by (x₀ + x₁)/2, (x₁ + x₂)/2, (x₂ + x₃)/2, and (x₃ + x₄)/2.

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Key Concepts

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

Subinterval Length

The subinterval length, denoted as ∆𝓍, is calculated by dividing the total length of the interval by the number of subintervals. In this case, the interval [1, 3] has a total length of 2 (3 - 1). With n = 4 subintervals, the length of each subinterval is ∆𝓍 = 2/4 = 0.5.

Grid Points

Grid points are the specific values that mark the boundaries of the subintervals within the partitioned interval. For the interval [1, 3] with n = 4 and ∆𝓍 = 0.5, the grid points are calculated as x₀ = 1, x₁ = 1.5, x₂ = 2, x₃ = 2.5, and x₄ = 3. These points help in evaluating Riemann sums.

Riemann Sums

Riemann sums are a method for approximating the area under a curve by summing the areas of rectangles formed over subintervals. The left Riemann sum uses the left endpoints of the subintervals (x₀, x₁, x₂, x₃), the right Riemann sum uses the right endpoints (x₁, x₂, x₃, x₄), and the midpoint Riemann sum uses the midpoints of each subinterval (1.25, 1.75, 2.25, 2.75).

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