Problem

Sketch the region enclosed by the given curves an…

06:42

Question

Answered step-by-step

Problem 19 Medium Difficulty

Sketch the region enclosed by the given curves and find its area.

$ y = \cos \pi x $ , $ y = 4x^2 - 1 $

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Sky Li

Numerade Educator

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

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 6

Applications of Integration

Section 1

Areas Between Curves

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 6

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

Video Transcript

So you need to sketch the region enclosed by those votarine. Curves say this is our x y play and so the first function is cosine pi x, that means x equals to plus n pi equals to negative 1 point. So have letter x equals to 1 half cosine packs equal to 0 and the x equals to 0 co. Sine 0 is pi, because it's 0 is 1 point right. This function should look like and this is a even function. So its symmetric by y axis looks like this: is x equals 1. This is negative. 1. Excuse me, this is x, equals to negative 1 and y equals to negative 1. Second, and the second curve is a problem: not x, equals 0. This curve across y equals 1 axe 0, like both negative 1. Across this point, and when y equal to 0, you know x equals to positive and negative 1 half, so it cross those 2 points in it isn't native half right. So this problem. I look like this and the enclosed region is in between this shaded area. So, in order to find this area, we write on integral form the area of integral with respect to retuse x, because those functions can be represented by x explicitly so this is 8 was to integral, with respect with respect to x and x goes from negative 1 Half to positive 1 half- and here is the upper curve minus the lower this cosine x, minus 4 x square minus 1 point. So it will be minus 4 x, squared plus 1, all right next, when you to evaluate this definite integral this definite integral. So, let's find anti derivative, the first term there will be 1 over pi sine pi x, and the second term will be 4 over 3 x cubed. The third time, just where we value at our half negative or half the first term at x, equals to half know sine pi over 2 is 1 is 1 over pi minus 4 over 3 plus 1 poi minus plugging x, equal to negative, but half. So, as you can see, those are odd functions. Therefore, everything flap their sign, so when the here is the first term will be negative. 1. Half negative, sorry mistake: we plug in x equals 1 half here, so this is not 4 over 3 times 1. This 4 over 3 times 1 over 8 point- and here is just a plus 1 half right and the second term just flap the sign o the second. This is minus 1 over pie and plus 43 times tone over 8 minus or half right, and since we have this negative sign, if we breed here, everything we'll put are, therefore we will get 2 times the first first terms. We only need to calculate the first term and are multiplied by 2 right, so here will be over 6 plus 1 half that's or 3, so you got 2 over pi, plus 2 thirds.

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