1. ## Shell Method Help

Here's the problem. I just have difficulty figuring out the problems where you are revolving a shape around something other than the x or y-axis. Any help would be greatly appreciated.

Here's the full problem:

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line.

$y=x^2$, $y=4x-x^2$, about the line $x=4$

2. Originally Posted by xfriendsonfirex
Here's the problem. I just have difficulty figuring out the problems where you are revolving a shape around something other than the x or y-axis. Any help would be greatly appreciated.

Here's the full problem:

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line.

$y=x^2$, $y=4x-x^2$, about the line $x=4$
$V = 2\pi \int_a^b \textcolor{red}{r(x)} \cdot \textcolor{blue}{h(x)} \, dx$

$V = 2\pi \int_0^2 \textcolor{red}{(4-x)}\textcolor{blue}{[(4x-x^2) - x^2]} \, dx$

3. Originally Posted by xfriendsonfirex
Here's the problem. I just have difficulty figuring out the problems where you are revolving a shape around something other than the x or y-axis. Any help would be greatly appreciated.

Here's the full problem:

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line.

$y=x^2$, $y=4x-x^2$, about the line $x=4$
the nice thing with the shell method is that the only major change happens on the radius. recall that for dx-problems, the radius is right - left.

the distance from the y-axis to the object is x. and your axis of rotation is given. just figure out which is on the right.

here, if you draw a diagram, you will realize that the axis of rotation is on the right of the region you are revolving. the axis is 4, the region is at a distance x, so the radius is right - left = (axis of rot.) - (distance from y-axis) = 4 - x

now do the Shell method with that radius.

4. Awesome, got it. Thanks!