Volumes of revolution help

Question

anonymous · Answer

Here's the question

Find the volume generated by revolving the region bounded by y = √x, y=2andx=0abouttheliney=2. Showallworkingusingexact values (do not use decimal approximations).

anonymous · Answer

The answer I got is 8pi, but I'm not sure whether what I did is correct or not

amistre64 · Answer

what did you did?

anonymous · Answer

8 PI

amistre64 · Answer

notice that the radius they are seeking is 2-sqrt(x)  after that, its just fun and games

anonymous · Answer

I did it with method of disks

anonymous · Answer

Why is it 2-sqrt x? I'll type up my working out

baldymcgee6 · Answer

amistre come help me?

amistre64 · Answer

the center of rotation is y=2; so think of this as going out 2 and then cutting off the sqrt(x) part to fit inside of the rotation

amistre64 · Answer

i aint got a mouse or id try to draw it out

amistre64 · Answer

maybe the touchpad will do it

amistre64 · Answer

|dw:1338948628815:dw|

anonymous · Answer

=$$\int\limits_{0}^{4} \pi[outer radius]^2 - \pi[innerradius]^2 $$
=$$\int\limits_{0}^{4} \pi[2]^2 - \pi[\sqrt(x)]^2 $$
=$$[(4*\pi*4)-(8\pi)]-[0-0)]$$
= 8pi

amistre64 · Answer

see how the radius is mathically; 2 - sqrt(x) at any given point?

amistre64 · Answer

that works too

amistre64 · Answer

or does it?

amistre64 · Answer

no, thats wrong

anonymous · Answer

I don't have any solutions or answers for it, so I'm not sure if I'm right or wrong =/

amistre64 · Answer

i see what you were thinking, and it wrong becasue your second part of it doesnt really match the situation correctly

amistre64 · Answer

pi (2-sqrt(x))^2; and that doesnt distribute like you have

amistre64 · Answer

the shell method would be a simple write up tho

anonymous · Answer

Oh I see, so the inner radius is the outer radius (2) take the function?

amistre64 · Answer

$$2pi\int_{0}^{2}y(y^2)dy$$

amistre64 · Answer

its the position of the spin axis that has to be taken into account and i just spun that sheel about y=0 lol ... oy i should try to sleep more

amistre64 · Answer

forget the shell, the disc is a fine method for this

anonymous · Answer

Ok, so the inner shell is (2-sqrtx)?

amistre64 · Answer

the inner radius of your disc; yes.  radius = 2-sqrt(x)

anonymous · Answer

Thank you very much

amistre64 · Answer

youre welome, just expand it out and its all power rules after that

amistre64 · Answer

baldy thinks i might have one more in me after that :)