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 5.2.75b

Chapter 5, Problem 5.2.75b

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₁⁴ 2√𝓍 d𝓍

Verified step by step guidance

Step 1: Understand the problem. The goal is to estimate the value of the definite integral ∫₁⁴ 2√𝓍 d𝓍 using midpoint Riemann sums with n = 20, 50, and 100. A midpoint Riemann sum approximates the integral by dividing the interval into n subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.

Step 2: Divide the interval [1, 4] into n subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / n. For n = 20, 50, and 100, compute Δ𝓍 for each case.

Step 3: Determine the midpoints of each subinterval. For each subinterval [𝓍ᵢ, 𝓍ᵢ₊₁], the midpoint is given by 𝓂ᵢ = (𝓍ᵢ + 𝓍ᵢ₊₁) / 2. Calculate the midpoints for all subintervals for n = 20, 50, and 100.

Step 4: Evaluate the function 2√𝓍 at each midpoint. For each midpoint 𝓂ᵢ, compute f(𝓂ᵢ) = 2√𝓂ᵢ. This gives the height of the rectangle for each subinterval.

Step 5: Compute the midpoint Riemann sum for each value of n. Multiply the function value at each midpoint by the width of the subinterval, Δ𝓍, and sum these products for all subintervals. This gives the approximate value of the integral for n = 20, 50, and 100.

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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 calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The notation ∫ₐᵇ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the accumulation of quantities.

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles, calculated using sample points within each subinterval, provides an estimate of the integral. The accuracy of the approximation improves as the number of rectangles (n) increases, making it essential to understand how to compute these sums effectively.

Midpoint Rule

The Midpoint Rule is a specific type of Riemann sum where the height of each rectangle is determined by the function value at the midpoint of each subinterval. This method often yields better approximations than using left or right endpoints, especially for functions that are continuous and smooth. By applying this rule with varying values of n, one can observe how the approximation converges to the actual value of the integral.

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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).

(b) Use geometry to find the displacement of the object between t = 0 and t = 2.

123

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Textbook Question

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₀¹ (𝓍² + 1) d𝓍

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Textbook Question

{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.

ƒ(𝓍) = 3 √x on [0,4] ; n = 40

(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.

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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.

(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)

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Textbook Question

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₀¹ cos ⁻¹ 𝓍 d𝓍

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Textbook Question

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

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.∫₁⁴ 2√𝓍 d𝓍

Verified video answer for a similar problem:

Key Concepts

Definite Integral

Riemann Sums

Midpoint Rule

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₁⁴ 2√𝓍 d𝓍