Day 6: Revolutionary Christmas Trees

Calculus:

Plotting the function gives half of a Christmas tree shape. Now we consider the layers of the Christmas tree when they are rotated.

The first layer from $-1 \leq y < 0$ gives a cylindrical stump of volume $\frac{\pi}{4}$ .

The second layer from $0 \leq y < 1$ joined with the fourth layer from $2 \leq y < 3$ gives a cone of volume $\frac{1}{3}(8 \pi)$ .

The third layer $1 \leq y < 2$ is a frustum of volume $\frac{1}{3}(\frac{13}{4}\pi)$ .

Adding these all together gives a total volume of $4 \pi$ , so $x = \frac{640 \pi}{4 \pi} = \boxed{160}$ .