Use the method of cylindrical shells to find the volume gen- erated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.

y = 4x + x^2 ; y = 3 ; about x = 1

Question

Use the method of cylindrical shells to find the volume gen- erated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.

y = 4x + x^2 ; y = 3 ;	about x = 1

zepdrix · Answer

`Sketch the region and a typical shell.`

Let's do this part first.
Drawing it out might make it easier to understand how to setup the integral.

aykayyy · Answer

I did.. and so far I have gotten V = integral(x=1 to x=3) of [2pi(x)(4x-x^2-3)]

idk if im just doing the math wrong or i set it up wrong...but i cant get it...the answer is 8pi/3

zepdrix · Answer

Oh boy I can't figure out what's going on with your numbers here :( Hmm

zepdrix · Answer

https://www.desmos.com/calculator/vhjceqdaxb Here is a graph of our region. We'll be integrating with respect to x from one point of intersection to the other. The points are clearly not x=1 and x=3. Where did those come from? <:o

zepdrix · Answer

Sorry that link didn't paste correctly the first time :)

aykayyy · Answer

sorry.... its actually y = 4x - x^2

aykayyy · Answer

it would be this now https://www.desmos.com/calculator/vhjceqdaxb

zepdrix · Answer

Oh.........

aykayyy · Answer

yeah.... so i set it up like this $$V = \int\limits_{1}^{3} 2\pi x (4x-x ^{2}-3)dx$$

zepdrix · Answer

Volume of a single shell would be,

V=(Circumference)(height)(Thickness)

You have the thickness correct, dx.
You have the height correct, upper function minus lower.

Your radius for the circumference is wrong.
The radius is x-1, since we're spinning around x=1.

That might fix things up for you :)

zepdrix · Answer

$$\large V = \int\limits\limits_{1}^{3} 2\pi (x-1) (4x-x ^{2}-3)dx$$

aykayyy · Answer

hahah i tried that as well.... and i got 89pi/3

hopefully i just did the algebra wrong... ill try again..

aykayyy · Answer

thanks for your help!

zepdrix · Answer

imma give it a run, see what i come up with :)

aykayyy · Answer

thanks :)

zepdrix · Answer

I hate working these out, it's so easy to make a mistake after you set it up :P

aykayyy · Answer

oh wow... i just did it and got 8pi/3 

it was a stupid algebra error... im just really tired... thanks anyways!

zepdrix · Answer

cool, good job c:

aykayyy · Answer

the algebra always kills me.... i put a plus instead of a minus or something all the time haha

zepdrix · Answer

heh