Volume of the solid using a cylindrical cross section

[Neutrinogun]

[STRIKE][/STRIKE]

Homework Statement

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Homework Equations

$y = \sqrt{x-1} , y = 0, x = 5;$ about $y = 3$

The Attempt at a Solution

I already completed graphing it, but not really sure how to show that here.

2$\pi$$\int_0^2$(shell radius)(shell height) $dy$
2$\pi$$\int_0^2$$(y)(y^2+1)$ $dy$
2$\pi$$\int_0^2$$(y^3+y)$ $dy$
$2\pi (((y^4)\div4)) + (y^2)\div2)$
$2\pi(4+2) - 0$
$12\pi$

Is that correct? I'm not sure if I did the shell height correctly.

 

[HallsofIvy]

You are rotating around y= 3 which means that the axis of each cylinder is y= 3 and the radius is 3- y, not y.

 

[Neutrinogun]

Aha, thank you so much! :)