Solution:
Let $T \subset \mathbb{R}^2$ denote the triangle, including its interior. Then $T$'s area is $5/2$, and its centroid is $(1004, 4/3)$, so
$$\int_{(x, y) \in T} x \, dx \, dy = \frac{5}{2} \cdot 1004 = 2510$$
We are interested in the volume
$$\int_{(x, y) \in T} 2\pi x \, dx \, dy$$
but this is just $2\pi \cdot 2510 = 5020\pi$.