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 the form of the definition of the integral gi…

06:14

Question

Answered step-by-step

Problem 24 Easy Difficulty

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.

$ \displaystyle \int^2_0 (2x - x^3) \, dx $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Robert Daugherty
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Robert Daugherty

Numerade Educator

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

More Answers

01:02

Frank Lin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 2

The Definite Integral

Related Topics

Integrals

Integration

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Grace He
Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

Video Thumbnail

40:35

Area Under Curves - Overview

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

Join Course
Recommended Videos

09:33

Use the form of the defini…

06:14

Use the form of the defini…

0:00

Use the form of the defini…

12:12

Use the form of the defini…

07:04

Use the form of the defini…

00:55

Use the form of the defini…

02:27

Use the form of the defini…

00:58

Use the form of the defini…

03:04

Use the form of the defini…

Watch More Solved Questions in Chapter 5

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
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

So problem 24, we're dealing with the limit definition of an integral. So a lot of algebra involved in solving these guys, I find it easiest just to break these up. So break some of the algebra into two separate integral. So two X dx Minour the integral from 0 to 2 of x cubed dx. Let's go do the work of each of those separate and then we'll come right back here to find our answer. So let's deal with the first integral. First the integral from 0 to 2. So the integral from 0 to 2 of two x. DX by definition is going to be too times the limit as N approaches infinity of the sun of I equals one to end. Now the width of each rectangle, it's just going to be to zero over in. So the width of each rectangle is too over in the height of each rectangle is going to be this function X evaluated so X at two over in Um four over in and and so forth. You're gonna increment this. So this is gonna be too I over in right? So this is the limit or it's going to be two times the limit as N approaches infinity. And if you look at what I have here, this is going to be you can replace that i with the formula that you know the some Mhm. I go one to end of I it's just simply end N Plus 1/2. So that's going to take the some of this out, the summation sign. So I'm gonna be left with two over in and then times two in Yeah, N plus one over two in. Yeah. So all I did was replace the what you see here in N plus 1/2, that replaces the I. So in this case, if I look at what I have to simplify this guy, this is two times the limit as N approaches infinity. And so what you see here is that you've got um to end over to end and then if you look at this, so this is going to be two and if I look at this in over N plus one, I can write that as one plus one over in. Mhm. And so as N goes to infinity, this term will go to zero. So this just becomes too um times two times one plus zero, which is simply four. So if I look at where I am in the scope of this problem, I have determined that the value of this one Is four. Okay, now I could have easily gotten that one graphically, you know, what is the line, you know, y equal to X From 0 to 2. Mhm. So when you plug in a two there you get four. Oh. Mhm. So right here, so if this is too yes and this is for the area that one half based sometimes should be four. So already got that. So I know that's right now. Little bit more work to get the integral. Yes, from 0 to 2 of X cubed dx. That is going to be the limit as in approaches infinity of the some I equal one to end with of each rectangle is still too over in and then now this is going to be um two. Um Yes, I cube so to I over in cute. So what is this going to give me? Well I know how to evaluate. Um let's just go and simplify this is the limit in approaches infinity of the sum I equal 12 N two cubed is eight. So this is going to be what, 16? So to to me so 16 Over into the 4th. I cubed. Ok. And so now we need our formula that the sum I equal one to end of I cubed is 1/4 in squared. Yeah, N plus one squared. Yeah. Mhm So let me get the some on this one real quick. This is going to be the limit. Yeah. Yeah. And approaches infinity. Uh So what have we got? 16 over? End of the fourth And then the sum of I cubed. It's just going to be 1/4 and then you've got in squared N plus one squared. So this is going to give me the limit and approaches infinity. 16/4 is four. And then I can write this as so you got in squared. This term will cancel if you write that as in square times in squared And then you can write this as one plus one over in he squared. And so this answer just turns out to be four. Go back to your original, So it's 4 -4. Final answer is zero.

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
67
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
45
Hosted by: Alonso M
See More

Related Topics

Integrals

Integration

Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

Video Thumbnail

40:35

Area Under Curves - Overview

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

Join Course
Recommended Videos

09:33

Use the form of the definition of the integral given in Theorem 4 to evaluate t…

06:14

Use the form of the definition of the integral given in Theorem 4 to evaluate t…

0:00

Use the form of the definition of the integral given in Theorem 4 to evaluate t…

12:12

Use the form of the definition of the integral given in Theorem 4 to evaluate t…

07:04

Use the form of the definition of the integral given in Theorem 4 to evaluate t…

00:55

Use the form of the definition of the integral given in Theorem 4 to evaluate …

02:27

Use the form of the definition of the integral given in the theorem to evaluate…

00:58

Use the form of the definition of the integral given in Theorem 4 to evaluate …

03:04

Use the form of the definition of the integral given in Theorem 4 to evaluate t…
Additional Mathematics Questions

03:16

Using Simpson's rule with 6 intervals, evaluate Jt Vi cos? dt Correct t…

03:32

A complete graph on n vertices Kn has 28 egdes. Find the following: (W) Numb…

04:30

34. In each part; find the determinant given that A is a 4 X 4 ma- trix for …

01:30

In a large university 13.5% of the.23 students take economics, 24.7% of the …

02:01

(a) Show that in a Banach space, an absolutely convergent series is converge…

01:07

which of the following is NOT an assumption of the Binomial distribution (ab…

03:51

(a) Suppose that f : X = Y is a continuous function and {xn} is a Cauchy seq…

04:04

(b) Let V be the vector space of all bounded or unbounded sequences of compl…

04:04

(b) Let V be the vector space of all bounded or unbounded sequences of compl…

00:01

Researchers collected information on the body parts of a new species of frog…

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
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started