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 5.1.41d
Chapter 5, Problem 5.1.41d

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


ฦ’(๐“) = 1/x on [1,6] ; n = 5


(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 ฦ’(๐“) = 1/x over the interval [1,6] with n = 5 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 [1,6] into n = 5 equal subintervals. The width of each subinterval, ฮ”๐“, is calculated as ฮ”๐“ = (6 - 1)/5 = 1. This means each subinterval has a width of 1.
Step 3: Determine the midpoints of each subinterval. The subintervals are [1,2], [2,3], [3,4], [4,5], and [5,6]. The midpoints are calculated as the average of the endpoints of each subinterval: (1+2)/2 = 1.5, (2+3)/2 = 2.5, (3+4)/2 = 3.5, (4+5)/2 = 4.5, and (5+6)/2 = 5.5.
Step 4: Evaluate the function ฦ’(๐“) = 1/x at each midpoint. Substitute each midpoint value into the function to find the heights of the rectangles: ฦ’(1.5), ฦ’(2.5), ฦ’(3.5), ฦ’(4.5), and ฦ’(5.5).
Step 5: Multiply each function value by the width of the subinterval, ฮ”๐“ = 1, and sum the results to calculate the midpoint Riemann sum. The formula is: Midpoint Riemann Sum = ฮ”๐“ * [ฦ’(1.5) + ฦ’(2.5) + ฦ’(3.5) + ฦ’(4.5) + ฦ’(5.5)].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
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 then 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.
Recommended video:
06:11
Introduction to Riemann Sums

Midpoint Rule

The midpoint rule is a specific type of Riemann sum that uses the midpoint of each subinterval to estimate the area under a curve. For a function f(x) over 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, provides 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 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 โˆซ_a^b f(x) dx and can be computed using various techniques, including Riemann sums.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (e) โˆซ d๐“/(81 + 9๐“ยฒ) (Hint: Factor a 9 out of the denominator first.)  

57
views
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       โˆ‘  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

79
views
Textbook Question

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

{Use of Tech} ฦ’(๐“) = e หฃ/โ‚‚ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums. 

119
views
Textbook Question

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

(d) If โˆซโ‚แต‡ ฦ’(๐“) d๐“ = โˆซโ‚แต‡ ฦ’(๐“) d๐“, then ฦ’ is a constant function. 

63
views
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.


โˆซโ‚€^ฯ€/2 cos ๐“ d๐“ ; n = 4

80
views
Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (f) โˆซ d๐“/โˆš36 โ€•๐“ยฒ

78
views