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Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.2 AIME Find the answer United States

Problem:
A triangle with vertices at $(1003,0)$ , $(1004,3)$ , and $(1005,1)$ in the $xy$ -plane is revolved all the way around the $y$ -axis. Find the volume of the solid thus obtained.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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 $2\pi \cdot 2510 = 5020\pi$ .

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