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 5.2.31c

Chapter 5, Problem 5.2.31c

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Verified step by step guidance

Step 1: Understand the problem. You are tasked with approximating the definite integral ∫₃⁶ (1 - 2𝓍) d𝓍 using left and right Riemann sums with n = 6. This means dividing the interval [3, 6] into 6 subintervals and calculating the sum of areas of rectangles using the left and right endpoints of each subinterval.

Step 2: Determine the width of each subinterval, Δ𝓍. The width is calculated as Δ𝓍 = (b - a) / n, where [a, b] is the interval of integration. Here, a = 3, b = 6, and n = 6. Substitute these values into the formula to find Δ𝓍.

Step 3: For the left Riemann sum, identify the left endpoints of each subinterval. These endpoints are x₀, x₁, ..., x₅, where x₀ = a and x₅ = b - Δ𝓍. Evaluate the function f(𝓍) = 1 - 2𝓍 at each left endpoint and multiply each value by Δ𝓍. Sum these products to approximate the integral using the left Riemann sum.

Step 4: For the right Riemann sum, identify the right endpoints of each subinterval. These endpoints are x₁, x₂, ..., x₆, where x₁ = a + Δ𝓍 and x₆ = b. Evaluate the function f(𝓍) = 1 - 2𝓍 at each right endpoint and multiply each value by Δ𝓍. Sum these products to approximate the integral using the right Riemann sum.

Step 5: Compare the left and right Riemann sums. These approximations provide an estimate of the definite integral ∫₃⁶ (1 - 2𝓍) d𝓍. The true value of the integral lies between these two sums, and the accuracy improves as n increases.

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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 a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by dividing the area under a curve into rectangles. The sum is calculated by taking the function's value at specific points (left endpoints, right endpoints, or midpoints) and multiplying by the width of the subintervals. As the number of rectangles increases, the Riemann sum approaches the exact value of the definite integral.

Left and Right Riemann Sums

Left and right Riemann sums are specific types of Riemann sums that use the leftmost and rightmost points of each subinterval, respectively, to determine the height of the rectangles. For 'n' subintervals, the left Riemann sum uses the function values at the left endpoints, while the right Riemann sum uses the values at the right endpoints. These sums provide different approximations of the definite integral, which can be compared for accuracy.

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Left, Right, & Midpoint Riemann Sums

Related Practice

Textbook Question

Working with area functions Consider the function ƒ and its graph.

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Textbook Question

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .

c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ?

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Textbook Question

Use Table 5.6 to evaluate the following definite integrals.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

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Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. (c) Calculate the left and right Riemann sums for the given value of n.∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Verified video answer for a similar problem:

Key Concepts

Definite Integral

Riemann Sum

Left and Right Riemann Sums

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6