Volumes by cylindrical shells

Question

Find the volume bound by $$xy=1;x=0;y=1;y=3; $$rotated about $x$-axis.

In my attempt, I determined that the shell height will be $1/y$ and the shell radius will be $y-1$

Clearly these are wrong as the solutions manual states the answer being $4\pi$

Answer 1

The radius of the shell is the distance from the slice being rotated to the axis of rotation; distance to the $x$-axis is $y$, not $y-1$.

Answer 2

Edit: I'm told that below is called the "washer" method, so probably not what you're looking for. I'll leave it up anyways and hopefully either OP or other future users can possibly benefit from it.

---

If you haven't already, I highly recommend drawing pictures for these types of problems. If you draw a picture, you can see there are two "pieces" that are being rotated. The first is a rectangle of width $\frac{1}{3}$ extending from $x = 0$ to $x = \frac{1}{3}$. The next part is bounded above by the graph $y = \frac{1}{x}$ and below by $y = 1$ from $x = \frac{1}{3}$ to $x = 1$.

Therefore, the first part of the solution will be to evaluate the following integrals:

$$\pi \int_0^{\frac{1}{3}} (3)^2 dx + \pi\int_{\frac{1}{3}}^1 \Big(\frac{1}{x}\Big)^2 dx$$

However, this is over-counting our volume since we are not subtracting off the volume contributed by the region below $y = 1$. Therefore, your final solution will be the following subtracted from the above result:

$$\pi \int_0^3 1 dx$$