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 5.1.31d

Chapter 5, Problem 5.1.31d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(𝓍) = e^(𝓍/2) over the interval [1, 4] with n = 6 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.

Step 2: Divide the interval [1, 4] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / 6 = 3 / 6 = 0.5.

Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function. The left endpoints are: 𝓍₀ = 1, 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, and 𝓍₅ = 3.5. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.

Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function. The right endpoints are: 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, 𝓍₅ = 3.5, and 𝓍₆ = 4. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.

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 5, and the right Riemann sum is Σ[ƒ(𝓍ᵢ) * Δ𝓍] for i = 1 to 6.

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Key Concepts

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

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 within these subintervals, and then summing the products of these values and the widths of the subintervals. The left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints.

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫[a,b] f(x) dx, where [a,b] is the interval of integration.

Function Evaluation

Function evaluation involves substituting specific values into a function to determine its output. In the context of Riemann sums, evaluating the function at the endpoints of the subintervals is crucial for calculating the sum. For the given function f(x) = e^(x/2), evaluating it at the left and right endpoints of each subinterval will provide the necessary function values to compute the left and right Riemann sums.

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Related Practice

Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

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Textbook Question

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.

(d) ∫ cos 𝓍/7 d𝓍

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then

B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

Textbook Question

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

{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4

(d) Calculate the left and right Riemann sums.