Revolution about horizontal and vertical lines

[cathy]

Homework Statement

Use shell method to find volume.
y=x+2
y=x^2
rotate about the x-axis




2. The attempt at a solution

I cannot seem to solve this. I thought this was the way to solve it, but I don't understand if I am missing something crucial.
This is how I set up the integral.

v=integral from 1 to 4 (2pi* y * (y-2-sqrt(y))
This is not giving me the correct answer. Is this that way to set this up?
I also tried the same integral from 0 to 4, and cannot determine what I am doing wrong.
I would appreciate any help.

 

[tiny-tim]

hi cathy! welcome to pf!

Use shell method to find volume.
…
v=integral from 1 to 4 (2pi* y * (y-2-sqrt(y))
This is not giving me the correct answer. Is this that way to set this up?
I also tried the same integral from 0 to 4, and cannot determine what I am doing wrong.

let's see … you're using cylindrical shells of radius y, thickness dy, and length x₂ - x₁

and that should be between where they meet, at (0,0) and at (2,4): ie for y between 0 and 4

so you second try should work …

can you show us your calculations?​

 

[cathy]

hello!
well, i tried the second one again, like you said, and here are my calculations:

taking the antideriv of,
y^2-2y-y^3/2
antideriv would be:
1/3y^3 - y^2 - 2/5y^5/2
plugging in 4, I get
64/3 -16 -64/5 = -112/15 *2pi
= -224/15pi
and that is not correct

Did I do something wrong here? Please advise.

 

[tiny-tim]

(just got up :zzz:)

looks ok (apart from everything being minus what it should be, since x² > x+2)

have you tried +224/15π ?

 

[cathy]

Ohh. i actually see the problem. since it's required for me to do shells, the problem needed to be divided into two separate integrals.
thank you for your help tiny-tim :)