Use the Shell Method to calculate the volume of rotation about the x-axis for the region underneath the graph.
$\displaystyle y=x^-1$ on interval [1,4]
Hi,
$\displaystyle V = 2\pi \int_0^1 y(y^-1) \, dy$
I set it up like that because if you solve for x of this: $\displaystyle y=x^-1=y$ you get $\displaystyle x=y^-1$.
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 $\displaystyle dy$, 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 ...
$\displaystyle 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
$