# Thread: Using Shell Method to calculate volume of rotation about x-axis

1. ## Using Shell Method to calculate volume of rotation about x-axis

Use the Shell Method to calculate the volume of rotation about the x-axis for the region underneath the graph.

$y=x^-1$ on interval [1,4]

2. Originally Posted by johnleyfield
Use the Shell Method to calculate the volume of rotation about the x-axis for the region underneath the graph.

$y=x^-1$ on interval [1,4]
show us what you've set up ...

3. Hi,

$V = 2\pi \int_0^1 y(y^-1) \, dy$

I set it up like that because if you solve for x of this: $y=x^-1=y$ you get $x=y^-1$.

Is that correct? I don't think it can be because you distribute the y in the integral you simply get y^0 which is just $dy$, correct? Please feel free to correct me because I am pretty sure my logic is wrong haha

4. Originally Posted by johnleyfield
Hi,

$V = 2\pi \int_0^1 y(y^-1) \, dy$

I set it up like that because if you solve for x of this: $y=x^-1=y$ you get $x=y^-1$.

Is that correct? I don't think it can be because you distribute the y in the integral you simply get y^0 which is just $dy$, correct? Please feel free to correct me because I am pretty sure my logic is wrong haha
look at the graph of the region (attached).

if you must use cylindrical shells, then you need to break it up into two integrals, because the length of your representative rectangles change ...

$2\pi \int_0^{\frac{1}{4}} y(4-1) \, dy + 2\pi \int_{\frac{1}{4}}^1 y\left(\frac{1}{y} - 1\right) \, dy
$