Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 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} Ζ’(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

Verified step by step guidance
1
Step 1: Understand the problem. The goal is to calculate the midpoint Riemann sum for the function Ζ’(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, Ξ”x, using the formula Ξ”x = (b - a) / n, where a = 1 and b = 3. This gives Ξ”x = (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 Ζ’(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, Ξ”x = 0.5, and sum the results to approximate the area under the curve. The midpoint Riemann sum is given by the formula: S = Ξ”x * [Ζ’(1.25) + Ζ’(1.75) + Ζ’(2.25) + Ζ’(2.75)].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

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.
Recommended video:
06:11
Introduction to Riemann Sums

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.
Recommended video:
Guided course
5:50
Power Rules

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.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

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

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.                                                                                                                                                

171
views
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 β‰₯ 5, find the function that gives the displacement between t = 0 and any time t β‰₯ 5.

59
views
Textbook Question

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) βˆ«β‚€^Ο€/¹⁢ sec Β² 4𝓍 d𝓍

80
views
Textbook Question

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

Ζ’(𝓍) = xΒ² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

95
views
Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) βˆ«β‚€Β³ 5Ζ’(𝓍) d𝓍

47
views
Textbook Question

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

{Use of Tech} Ζ’(𝓍) = cos 𝓍 on [0. Ο€/2]; n = 4

(d) Calculate the left and right Riemann sums.

109
views