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 5.1.39d

Chapter 5, Problem 5.1.39d

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

{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4

(d) Calculate the midpoint Riemann sum.

Verified step by step guidance

Step 1: Understand the problem. The goal is to calculate the midpoint Riemann sum for the function ƒ(𝓍) = √x over the interval [1, 3] with n = 4 subintervals. A midpoint Riemann sum approximates the area under the curve by summing the areas of rectangles whose heights are determined by the function value at the midpoint of each subinterval.

Step 2: Divide the interval [1, 3] into n = 4 equal subintervals. To do this, calculate the width of each subinterval, Δx, using the formula Δx = (b - a) / n, where a = 1 and b = 3. This gives Δx = (3 - 1) / 4 = 0.5.

Step 3: Determine the midpoints of each subinterval. The subintervals are [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. The midpoints are calculated as the average of the endpoints of each subinterval: (1 + 1.5)/2 = 1.25, (1.5 + 2)/2 = 1.75, (2 + 2.5)/2 = 2.25, and (2.5 + 3)/2 = 2.75.

Step 4: Evaluate the function ƒ(𝓍) = √x at each midpoint. This means calculating √1.25, √1.75, √2.25, and √2.75. These values represent the heights of the rectangles for the Riemann sum.

Step 5: Multiply each function value by the width of the subinterval, Δx = 0.5, and sum the results to approximate the area under the curve. The midpoint Riemann sum is given by the formula: S = Δx * [ƒ(1.25) + ƒ(1.75) + ƒ(2.25) + ƒ(2.75)].

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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 summing the products of these values and the widths of the subintervals. The midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function, providing a more accurate approximation than using the endpoints.

Midpoint Rule

The midpoint rule is a specific type of Riemann sum that estimates the area under a curve by taking the value of the function at the midpoint of each subinterval. For an interval [a, b] divided into n equal parts, the midpoint of each subinterval is calculated, and the function is evaluated at these midpoints. The sum of these values, multiplied by the width of the subintervals, gives the midpoint Riemann sum, which serves as an approximation of the integral.

Definite Integral

The definite integral of a function over an interval [a, b] represents the net area under the curve of the function between the two points a and b. It is a fundamental concept in calculus, linking the concepts of area and accumulation. The definite integral can be computed using various methods, including Riemann sums, and is denoted as ∫_a^b f(x) dx, where f(x) is the function being integrated.

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Related Practice

Textbook Question

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.

Textbook Question

Use Table 5.6 to evaluate the following definite integrals.

(d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

Textbook Question

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

ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums.

Textbook Question

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

(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function.

Textbook Question

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

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..

∫₀² (𝓍²―2) d𝓍 ; n = 4

Textbook Question

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

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4(d) Calculate the midpoint Riemann sum.

Verified video answer for a similar problem:

Key Concepts

Riemann Sums

Midpoint Rule

Definite Integral

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

{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4

(d) Calculate the midpoint Riemann sum.