6. A triangle with vertices at $(1003,0),(1004,3)$, and $(1005,1)$ in the $x y$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained.
Answer: $5020 \pi$
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
$$
\int_{(x, y) \in T} x d x d y=\frac{5}{2} \cdot 1004=2510 .
$$

We are interested in the volume
$$
\int_{(x, y) \in T} 2 \pi x d x d y,
$$
but this is just $2 \pi \cdot 2510=5020 \pi$.