Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Use (a) the Trapezoidal Rule, (b) the Midpoint Ru…

00:30

Question

Answered step-by-step

Problem 17 Easy Difficulty

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of $ n $. (Round your answers to six decimal places.)

$ \displaystyle \int_0^4 \ln (1 + e^x)\ dx $ , $ n = 8 $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Chris Trentman
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Chris Trentman

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

00:28

Yuou Sun

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Related Topics

Integration Techniques

Discussion

You must be signed in to discuss.
CB

Claire B.

October 19, 2021

Sorry, I need to elaborate: how would you use the TI-84 to perform Simpson's Rule for something as high as n=50?

CB

Claire B.

October 19, 2021

Simpson's rule with n=50

Top Calculus 2 / BC Educators
Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

12:43

Use (a) the Trapezoidal Ru…

12:16

Use (a) the Trapezoidal Ru…

00:46

Use (a) the Trapezoidal Ru…

11:21

Use (a) the Trapezoidal Ru…

12:47

Use (a) the Trapezoidal Ru…

00:36

Use (a) the Trapezoidal Ru…

07:11

Use (a) the Trapezoidal Ru…

12:12

Use (a) the Trapezoidal Ru…

07:41

Use (a) the Trapezoidal Ru…

03:41

Use (a) the Trapezoidal Ru…

14:42

Use (a) the Trapezoidal Ru…

12:45

Use (a) the Trapezoidal Ru…

06:51

Use (a) the Trapezoidal Ru…

00:36

Use (a) the Trapezoidal Ru…

12:03

Use (a) the Trapezoidal Ru…

Watch More Solved Questions in Chapter 7

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50

Video Transcript

were given an integral in the value of n and in each part where asked to use a certain rule toe. Approximate this interval with this value event. Okay, the integral is the integral from 0 to 4 of the natural log of one plus e to the X. I mean where n is equal to eight in part a were asked to use the trapezoidal rule. So first for this rule you want to determine the interval lengths Delta X These are be right most endpoint four minus the left most endpoint zero over n, which is eight. This is 4/8 or in a decimal notation 0.5. The next step is to find required function values. So in trapezoidal rule, we want to find the values of the function on the end points of these sub intervals. So I'll make a table on the left side, we have the end points X and on the right side we have the value of our function, which is natural log of one plus e to the X mhm. Let's start off are left endpoint zero and will increase by Delta X or 0.5 until we reach four. So we have zero 0.51 point 1.5 and so on. Mhm. Now, for the right side, all we have to do is simply plug in attention X from the left side. So, for example, plugging in X equals zero. You have the natural log of to for our first entry in decimal notation, this is to eight decimal places 0.693 14718 Likewise, we plug in X equal 2.5. We get approximately 0.974076 98 We continue in this fashion all the way to X equals three. And so your table of values should look something like this paying and how accurate want to be. You may have more or less decimal places on the right. Now that we have these values, we'll use the formula for the trapezoidal rule for approximation, which will call T eight for trapezoidal. We know this has n plus one or a total of nine terms, and it's some and that t eight is equal to our formula is the in the interval length felt x over two times f of the first point x zero So this is zero plus two times ffx, one plus two times ffx, too. And we continue in this fashion all the way up to two times ffx seven and then plus just one times ffx eight, which is three. And then we simply plug in the values for Delta X, which is 0.5, and all of our values from the table into the some in the answer is approximately 8.804227 to six decimal places. That's the answer for Party now in Part B were asked to use the midpoint rule. This is similar to party, except for in the mid point rule. We have the same steps size Delta X equals five, but now we want to find the values of the function natural log of one plus e. D. X at the midpoint of these intervals. So we're going to start drawing another table. First column X second column is the natural lot of one plus e to the X. Yeah, and instead of starting with X equals zero, the left endpoint we're going to start at the midpoint of the first sub interval, 0.5, which is 0.25 Then we're going to add five until we reached the midpoint of 3.540 which is 3.75 So we have 25 plus 250.5 point 75 plus five is. Mm hmm. 1.25 plus 0.5 is 1.75 And so on, then, just as in part A, we're going to plug in the values and left to get the values on the right. So, for example, plugging an X equals 0.25 and two natural log of one plus e to the X. This is approximately 0.825939 42 However many decimal places you want, I would suggest a least six. Likewise, we plug in 0.75 We get 1.136 871 and I'll continue this all the way down the table. Table of values should look something like this. And finally, we're going to use the formula. So we know that the midpoint rule our approximation will call M eight. We know that this is going toe, have the total of n or eight terms in its some and the formula is the N eight is the length of a sub interval Delta X, which is 0.5 times the values of our function at the midpoint of each of these intervals. So we have from the table above f of x zero plus f of x one plus all the way up to F FX seven. So we have a total of eight terms in our son. And if you plug this into your calculator, this is approximately 8.799 212 to six decimal places under part C, whereas to use Simpson's rule approximately interval integral I mean now Simpsons World actually pretty similar to the trapezoidal rules. I'm going to overwrite heart a here, so this now becomes part C just is in part a and part B. The interval length is 25 and also just is in part a. The table is going to consist of on the left all the end points of the sub intervals on the right, the value the functions at those end points. So this table is the same as in part egg. But now our formula is going to be a little bit different, so below party here, I'll finish part C. So we know that s a r approximation giving substance rule as n plus one for nine terms just like part A and nine terms. But we know that these form of S eight is going to be the some interval with Delta X over three now instead of two times f of x zero plus four times value of after the next and point X one plus two times the value that for the next 10 point at thanks to plus four times that I have effort the next endpoint x three. We continue in this fashion, alternating between four and two until we get all the way up to four times half of x eight. Sorry, it should be four times f of x seven plus two times epic Lexi. Well, it's not really four times after that. Seven since sevens on. And then just one time different X eight puts the last turn. It's a total of nine terms in ourselves. And so if you plug this and do a calculator using the values from our first table up here, this is approximately 8.804 229 out to six decimal places. Okay,

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
153
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

12:43

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

12:16

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

00:46

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

11:21

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

12:47

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

00:36

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

07:11

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

12:12

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

07:41

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

03:41

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

14:42

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

12:45

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

06:51

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

00:36

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …

12:03

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to …
Additional Mathematics Questions

06:13

A dart is thrown upward with an initial velocity of 58 ftlsec from …

02:02


a. Females have a statistically significantly lower average
total cha…

03:58

5 points) Let X be a discrete random variable with P(X = i) = i/c f…

01:53

Upvote for correct, fast and
detailed answer

The number of cars c…

03:33

also how to write the solution for A and b in a fraction
b. Let event A =…

01:55

Annie is concerned over report that a woman over age 40 has better …

05:17

2 Delegates from four different countries are attending an arms mee…

01:44

There are dozens of personality tests available on the internet; On…

03:17

Suppose that you are offered the following "deal." You roll a six …

07:28

A recent study of the laundry habits of Americans included the time…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started