Problem

The Second Theorem of Pappus is in the same spiri…

03:26

Question

Answered step-by-step

Problem 47 Hard Difficulty

The centroid of a $ curve $ can be found by a process similar to the one we used for finding the centroid of a region. If $ C $ is a curve with length $ L $, then the centroid is $ (\bar{x}, \bar{y}) $ where $ \bar{x} = (\frac{1}{L}) \int x\ ds $ and $ \bar{y} = (\frac{1}{L}) \int y\ ds $. Here we assign appropriate limits of integration, and $ ds $ is as defined in Sections 8.1 and 8.2. ( The centroid often doesn't lie on the curve itself. If the curve were made of wire and placed on a weightless board, the centroid would be the balance point on the board.) Find the centroid of the quarter-circle $ y = \sqrt{16 - x^2} $, $ 0 \le x \le 4 $.

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Doruk Isik

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 8

Further Applications of Integration

Section 3

Applications to Physics and Engineering

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 8

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

Video Transcript

Hey, miss problem. We're giving any question of any 1/4 for circle. Why? Didn't give any question is why's it called scourge sexy sex card. If he noted exchanges between 04 and thes e questions are also known and given in a problem statement, we're trying to come play export and white bar for a 1/4 off a circle. All right, since we're going to use be given equations, we first need to calculate D s. And this is our be calculated in order Chocolate D s. We would need to first find the thereto. Ought to give any question which respect him. Thanks. So let's tag narrative off. Why would suspect Activity X would be won over Skirt one over to skirted of 16 lines, expiry months. Bye bye. There was intervention. It's native to X Tuesday against love. So we have that one has negative X delight. I'm skirted of 16 months excrement and skirt of that S O D I D X skirt would done be equal to X squared. Divided by 16 months. Ex skirt. All right, we have everything we need. Less display goes in. Export is equal to one over l what is L l in this case is the circumference. And for a circle circumference is two point times radius. And as you can see, Radius s four side will be to buy times for I've ever given 1/4 of the circles of in to divide this spot for So does your conference off that peace will only be two ply or s o for expert. We have won over length one or two by integral forms here before x times d s where d s is scribbled off one plus thes y d X skirt that is equal to x Great delight by 16 lines Expert The X that is equal to one over to pie trickle from 0 to 4 X times skirt of 16 divided by a skirt of 16 months Ex crime DX How did I get this Second port? Well, all I did was to multiplied this, uh, one by 16 minus explorer. And do I buy the same number? Skirted off 16 is equal to four. And we can just take this one outside the in. Jekyll and right export as, um four over to pie. Internal dessert for ex skirted off 16 months Ex card here, We're going to say that 16 months Next. Carter's our new variable. You saw that native to X E X would be to you and X t x Could dummy continue native d'you or two? And since dishonorable one X is zero you 16 and one access for you. The zeroes are the limits of formula In title will be then from 16 too zero he have negative d'you deal. I skirted a few soling Thus we find expert to be equal to eight or pie LSD chocolate white bar Why bar is worn over el again one over to buy integrate forms here to walk For this time we're gonna let's buy Why buy deep s we know what dances and we know why as a function of X status given to us So that'll be 16 months Ex crane multiplied by, uh skirted up d s, which is this part in which we calculated it to be as this term. So that would be skirted of 16. Delighted by skirted off 16 minus expert the ex Nazis will cancel out skirted off 16. It's four. So we have four over to pie off ex were exchanged between 04 That is an equal to 16 over to pie, which is eight or apply. So from this, we can export to be equal to eight over five. Why part to be equal to eight over pi.

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

The Second Theorem of Pappus is in the same spiri…

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

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Heather Zimmers

Kayleah Tsai

Caleb Elmore

Joseph Lentino

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 8

Video Transcript

James Stewart

Calculus: Early Transcendentals

Heather Zimmers

Kayleah Tsai

Caleb Elmore

Joseph Lentino

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

Calculus: Early Transcendentals

Heather Zimmers

Kayleah Tsai

Caleb Elmore

Joseph Lentino

James StewartCalculus: Early Transcendentals

Heather Zimmers

Kayleah Tsai

Caleb Elmore

Joseph Lentino

James Stewart

Calculus: Early Transcendentals