Question

sorry for asking again, when should i use disc method and shell method when i need to find the volume by integration?

Answer 1

whenever you feel that one is easier than the other to perform

Answer 2

let say, the question is:
Find the volume of the solid of revolution generated by revolving the region bounded by the curve y=xlnx, the x axis and the line x=1 and x=3 about the y-axis. what should i do?

Answer 3

draw a picture :)

Answer 4

\[\int_{1}^{3}2pi(xlnx)dx\]

might be simplest

Answer 5

forgot an x in that ...

Answer 6

2pix * f(x)

Answer 7

or if es are better for you; convert it to es and play along

Answer 8

so you use shell method, but why?

Answer 9

becasue it works?

Answer 10

what would y=x lnx look like if you solve for x

Answer 11

y = ln x^x

e^y = x^x realy isnt a simple function to play with

Answer 12

so the shel method would be your simplest route to take

Answer 13

so the curve y=xlnx revolved about y axis = use shell method?

Answer 14

it equals: use the easiest method. That method just so happens to be the shell, yes

Answer 15

a disc and a shell is really similar..

Answer 16

they are similar, except for the way the process looks at the function

Answer 17

the disc in this case looks at x(y)

the shell in this case looks at y(x)

Answer 18

x(y) is pretyy hard to define when y(x) = xlnx

Answer 19

x(y) =?

Answer 20

dunno, since e^y = x^x we cant really get an explicit form for x(y)

Answer 21

What does x(y) mean?

Answer 22

the same thing any function notation means; x as a function of y

Answer 23

y(x) means y as a function of x

r(t) means r as a function of t

etc .....

Answer 24

x(y) in this case would need to be the inverse of y(x)

Answer 25

so if it can be easily expressed in terms of y, disc method can also be used?

Answer 26

yes, if it could be easily expresed in terms of y, then disc would be fine to use

Answer 27

the method doesnt change the value

Answer 28

which is more; 5 gallons of water in a blue vase; or 5 gallons of water in a purple vase?

Answer 29

blue vase? purple case?

Answer 30

express the equation into an easier form to integrate

when y is expressed in terms of x, and it is revolved about y axis, shell method is used

when x is expressed in terms of y, and it is revolved about y axis, disc method is used

can i conclude in this way?

Answer 31

in general, no

Answer 32

the disc method is blocked by this problem simply becasue there is no way to define x(y) in a usable maner to play with the disc method

Answer 33

the only option we have is to use the shell method

Answer 34

then how would you determine which is the eaiser way?

Answer 35

easier

Answer 36

http://www.wolframalpha.com/input/?i=inverse+y%3Dx*lnx

that W function they use aint something that can be integrated with the stuff we are taught

Answer 37

i think i understand it a little, thanks :)