Suppose that the rectangular sheet of paper has dimensions $a$ and $b$, with $b$ being the longer side. We calculate the volume of the two cylinders formed by joining the long sides and short sides, respectively.

* Suppose the short sides are glued together. Then the height of the cylinder is $a$ and the circumference of the cylinder is $b$. If $r$ is the radius of the cylinder, it means that $2\pi r = b$, or $r = \frac{b}{2\pi}$. Hence the cylinder's volume is $\pi r^2 h = \pi (\frac{b}{2\pi})^2 \cdot a = \frac{ab^2}{4\pi}$.

* Suppose the long sides are glued together. Then the height of the cylinder is $b$ and the circumference of the cylinder is $a$. If $r$ is the radius of the cylinder, it means that $2\pi r = a$, or $r = \frac{a}{2\pi}$. Hence the cylinder's volume is $\pi r^2 h = \pi (\frac{a}{2\pi})^2 \cdot b = \frac{a^2b}{4\pi}$.

Comparing these two numbers, we see that the first volume can be written as $b(\frac{ab}{4\pi})$, while the second volume is $a(\frac{ab}{4\pi})$. Since the number in brackets is the same for both and $b > a$, it means that the first volume $b(\frac{ab}{4\pi})$ is the largest. Hence the largest volume is obtained when the short sides of the sheet of paper are glued together.