Problem

Use Property 8 to estimate the value of the integ…

00:28

Question

Answered step-by-step

Problem 59 Easy Difficulty

Use Property 8 to estimate the value of the integral.

$ \displaystyle \int^1_0 x^3 \,dx $

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Yuki Hotta

Numerade Educator

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

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

Well, let's go ahead and answer this question. It says the integral off X cubed between 0 to 1. It has to be between zero and one inclusive. Prove this using property eight. Okay, It's actually quite intuitive group. Basically, what you have to say is, if you have a new integration from A to B of ffx, it is always going to be between these two values M times B minus A and large and times B minus A When F of X is between large m and small m. What do they mean by that? Well, let's pretend that ffx looks something like this okay between A and B. Let's say that the lower bound of this is M and M doesn't necessarily be the point at f of A as just as long as F of X is, um, larger than small. M n e m would satisfy. Let's say that the upper bound iss large m So right now you see that f of X is between small M and large m. So what is M times B minus a equal to well, this portion is the area under this curve off em. So it's this red rectangle. Similarly, if I take a look at this value, it's the area off the blue rectangle. So what can I say about the area off the integral? The area under this curve? Well, it's very clear that it's going to be between or larger than the blue rectangle, but smaller than the red rectangle. So that's what Property eight is talking about. Okay, so all you need to do to show this is that the rectangle, the red rectangle, is one. The blue rectangle is zero. Why did I know that X cubed looks like this crime 01 when X is equal to zero, the left end. It's right here. So X cubed is greater than or equal to zero. Is it less than or equal to one? Yeah. If I plug in X is equal to one. The height is one X cubed is less than one. So that distance from 0 to 1 is one minus zero, which is equal to one. So to be a little bit careful here, I can say that this is zero times one minus zero, less than or equal to the area between zero toe one of X cubed the X and one times one minus zero, which basically represents the area under the red rectangle. The area under the blue rectangle which is sandwiching the area off. Ex cute. And this is how you prove this statement using Property eight.

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.

Problem